Lamina comprising cube corner elements and retroreflective sheeting

ABSTRACT

The present invention is directed to lamina(e) comprising cube corner elements, a tool comprising an assembly of laminae and replicas thereof. The invention further relates to retroreflective sheeting.

RELATED APPLICATIONS

[0001] This application claim priority to provisional U.S. PatentApplication Serial No. 60/452464 filed Mar. 6, 2003

FIELD OF THE INVENTION

[0002] The present invention is directed to a lamina comprising cubecorner elements, a tool comprising an assembly of laminae andreplications thereof including in particular retroreflective sheeting.

BACKGROUND OF THE INVENTION

[0003] Retroreflective materials are characterized by the ability toredirect light incident on the material back toward the originatinglight source. This property has led to the widespread use ofretroreflective sheeting for a variety of traffic and personal safetyuses. Retroreflective sheeting is commonly employed in a variety ofarticles, for example, road signs, barricades, license plates, pavementmarkers and marking tape, as well as retroreflective tapes for vehiclesand clothing.

[0004] Two known types of retroreflective sheeting are microsphere-basedsheeting and cube corner sheeting. Microsphere-based sheeting, sometimesreferred to as “beaded” sheeting, employs a multitude of microspherestypically at least partially embedded in a binder layer and havingassociated specular or diffuse reflecting materials (e.g., pigmentparticles, metal flakes or vapor coats, etc.) to retroreflect incidentlight. Due to the symmetrical geometry of beaded retroreflectors,microsphere based sheeting exhibits the same total light returnregardless of orientation, i.e. when rotated about an axis normal to thesurface of the sheeting. Thus, such microsphere-based sheeting has arelatively low sensitivity to the orientation at which the sheeting isplaced on a surface. In general, however, such sheeting has a lowerretroreflective efficiency than cube corner sheeting.

[0005] Cube corner retroreflective sheeting typically comprises a thintransparent layer having a substantially planar front surface and a rearstructured surface comprising a plurality of geometric structures, someor all of which include three reflective faces configured as a cubecorner element.

[0006] Cube corner retroreflective sheeting is commonly produced byfirst manufacturing a master mold that has a structured surface, suchstructured surface corresponding either to the desired cube cornerelement geometry in the finished sheeting or to a negative (inverted)copy thereof, depending upon whether the finished sheeting is to havecube corner pyramids or cube corner cavities (or both). The mold is thenreplicated using any suitable technique such as conventional nickelelectroforming to produce tooling for forming cube cornerretroreflective sheeting by processes such as embossing, extruding, orcast-and-curing. U.S. Pat. No. 5,156,863 (Pricone et al.) provides anillustrative overview of a process for forming tooling used in themanufacture of cube corner retroreflective sheeting. Known methods formanufacturing the master mold include pin-bundling techniques, directmachining techniques, and techniques that employ laminae.

[0007] In pin bundling techniques, a plurality of pins, each having ageometric shape such as a cube corner element on one end, are assembledtogether to form a master mold. U.S. Pat. No. 1,591,572 (Stimson) andU.S. Pat. No. 3,926,402 (Heenan) provide illustrative examples. Pinbundling offers the ability to manufacture a wide variety of cube cornergeometries in a single mold, because each pin is individually machined.However, such techniques are impractical for making small cube cornerelements (e.g. those having a cube height less than about 1 millimeter)because of the large number of pins and the diminishing size thereofrequired to be precisely machined and then arranged in a bundle to formthe mold.

[0008] In direct machining techniques, a series of grooves are formed inthe surface of a planar substrate (e.g. metal plate) to form a mastermold comprising truncated cube corner elements. In one well knowntechnique, three sets of parallel grooves intersect each other at 60degree included angles to form an array of cube corner elements, eachhaving an equilateral base triangle (see U.S. Pat. No. 3,712,706(Stamm)). In another technique, two sets of grooves intersect each otherat an angle greater than 60 degrees and a third set of groovesintersects each of the other two sets at an angle less than 60 degreesto form an array of canted cube corner element matched pairs (see U.S.Pat. No. 4,588,258 (Hoopman)). In direct machining, a large number ofindividual faces are typically formed along the same groove formed bycontinuous motion of a cutting tool. Thus, such individual facesmaintain their alignment throughout the mold fabrication procedure. Forthis reason, direct machining techniques offer the ability to accuratelymachine very small cube corner elements. A drawback to direct machiningtechniques, however, has been reduced design flexibility in the types ofcube corner geometries that can be produced, which in turn affects thetotal light return.

[0009] In techniques that employ laminae, a plurality of thin sheets(i.e. plates) referred to as laminae having geometric shapes formed onone longitudinal edge, are assembled to form a master mold. Techniquesthat employ laminae are generally less labor intensive than pin bundlingtechniques because fewer parts are separately machined. For example, onelamina can typically have about 400-1000 individual cube cornerelements, in comparison to each pin having only a single cube cornerelement. However, techniques employing laminae have less designflexibility in comparison to that achievable by pin bundling.Illustrative examples of techniques that employ laminae can be found inEP 0 844 056 A1 (Mimura et al.); U.S. Pat. No. 6,015,214 (Heenan etal.); U.S. Pat. No. 5,981,032 (Smith); and U.S. Pat. No. 6,257,860(Luttrell).

[0010] The base edges of adjacent cube corner elements of truncated cubecorner arrays are typically coplanar. Other cube corner elementstructures, described as “full cubes” or “preferred geometry (PG) cubecorner elements”, typically comprise at least two non-dihedral edgesthat are not coplanar. Such structures typically exhibit a higher totallight return in comparison to truncated cube corner elements. Certain PGcube corner elements may be fabricated via direct machining of asequence of substrates, as described in WO 00/60385. However, it isdifficult to maintain geometric accuracy with this multi-stepfabrication process. Design constraints may also be evident in theresulting PG cube corner elements and/or arrangement of elements. Bycontrast, pin bundling and techniques that employ laminae allow for theformation of a variety of shapes and arrangements of PG cube cornerelements. Unlike pin bundling, however, techniques that employ laminaealso advantageously provide the ability to form relatively smaller PGcube corner elements.

[0011] The symmetry axis of a cube corner is a vector that trisects thestructure, forming an equal angle with all three cube faces. In theaforementioned truncated cubes of Stamm, the symmetry axis is normal tothe equilateral base triangle and the cubes are considered to have nocant or tilt. The nomenclature “forward canting” or “positive canting”has been used in the cube corner arts to describe truncated cube cornerelements canted in a manner that increases only one base triangleincluded angle relative to 60°. Conversely, the nomenclature “backwardcanting” or “negative canting” has been used in the cube corner arts todescribe cube corner elements canted in a manner that increases two ofthe included angles of the base triangle relative to 60°. See U.S. Pat.No. 5,565,151 (Nilsen) and U.S. Pat. No. 4,588,258 (Hoopman). Canting ofPG cube corner elements is described in U.S. Pat. No. 6,015,214 (Heenanet al.).

[0012] Canting cube corner elements either backward or forward enhancesentrance angularity. Full cube corner elements have a higher total lightreturn than truncated cube corner elements for a given amount of cant,but the full cubes lose total light return more rapidly at higherentrance angles. One benefit of full cube corner elements is highertotal light return at low entrance angles, without substantial loss inperformance at higher entrance angles.

[0013] A common method for improving the uniformity of total lightreturn (TLR) with respect to orientation is tiling, i.e. placing amultiplicity of small tooling sections in more than one orientation inthe final production, as described for example in U.S. Pat. No.4,243,618 (Van Arnam), U.S. Pat. No. 4,202,600; and U.S. Pat. No.5,936,770 (Nestegard et al.). Tiling can be visually objectionable.Further, tiling increases the number of manufacturing steps in makingthe tooling employed for manufacture of the sheeting.

[0014] In addition to being concerned with the TLR, the performance ofretroreflective sheeting also relates to the observation angularity ordivergence profile of the sheeting. This pertains to the spread of theretroreflected light relative to the source, i.e. typically, vehicleheadlights. The spread of retroreflected light from cube corners isdominated by effects including diffraction, polarization, andnon-orthogonality. For this purpose, it is common to introduce angleerrors such as described in Table 1 of column 5 of U.S. Pat. No.5,138,488 (Szczech).

[0015] Similarly, Example 1 of EP 0 844 056 A1 (Mimura) describes a flycutting process in which the bottom angles of V-shaped grooves formedwith a diamond cutting tool were slightly varied in regular order, threetypes of symmetrical V-shaped grooves having depths of 70.6 μm, 70.7 μmand 70.9 μm were successively and repeatedly cut at a repeating pitch of141.4 μm in a direction perpendicular to the major surfaces of thesheets. Thus, a series of successive roof-shaped projections havingthree different vertical angles of 89.90°, 90.0°, and 91.0° in arepeating pattern were formed on one edge of the sheets.

[0016] Although the art describes a variety of retroreflective designsand their measured or calculated retroreflective performance; industrywould find advantage in retroreflective sheeting having new cube corneroptical designs and methods of manufacturing, particularly thosefeatures that contribute to improved performance and/or improvedmanufacturing efficiencies.

SUMMARY OF THE INVENTION

[0017] In one embodiment, the invention discloses a lamina comprisingcube corner elements having faces formed from grooves wherein adjacentgrooves range from being nominally parallel to nonparallel by less than1°. The adjacent grooves have included angles that differ by at least2°. In one aspect the included angles of the grooves are arranged in arepeating pattern. In another aspect, the faces of the elementsintersect at a common peak height. In yet another aspect, the grooveshave bisector planes that range from being mutually nominally parallelto nonparallel by less than 1°.

[0018] In another embodiment, the invention discloses a laminacomprising preferred geometry cube corner elements wherein at least aportion of the cube corner elements are canted having an alignment angleselected from alignment angles between 45° and 135°, alignment anglesbetween 225° and 315°, and combinations thereof. Preferably, a firstcube corner element is canted having an alignment angle between 60° and120° and a second adjacent cube is canted having an alignment anglesbetween 240° and 300°. Further, the alignment angle of the first cubepreferably differs from 0° or 180° by substantially the same amount asthe alignment angle of the second cube differs.

[0019] In each of these embodiments, the cube corner elements preferablycomprise faces formed from alternating pairs of side grooves. Theincluded angle of each pair of side grooves preferably has a sum ofsubstantially 180°. Further, the included angle of a first groove ispreferably greater than 90° by an amount of at least about 5° (e.g.about 10° to about 20°) and the included angle of a second adjacentgroove is less than 90° by about the same amount.

[0020] In another embodiment, the invention discloses a lamina having amicrostructured surface comprising cube corner elements having facesformed from a side groove set wherein at least two grooves within theset are nonparallel by amounts ranging from greater than nominallyparallel to about 1°. The elements preferably comprise dihedral angleerrors having magnitudes between 1 arc minute and 60 arc minutes. Thedihedral angle errors are preferably arranged in a repeating pattern.The grooves comprise skew and/or inclination that vary in sign and ormagnitude.

[0021] In all disclosed embodiments, the adjacent grooves are preferablyside grooves. Further, the elements preferably each have a face in acommon plane that defines a primary groove face. In addition, theelements are preferred geometry cube corner elements.

[0022] In other embodiments, the invention discloses a master toolcomprising a plurality of any one or combination of described lamina.The laminae are preferably assembled such that cube corner elements ofadjacent laminae are in opposing orientations. The elements preferablyhave a shape in plan view selected from trapezoids, rectangles,parallelograms, pentagons, and hexagons.

[0023] In other embodiments, the invention discloses replicas of themaster tool including multigenerational tooling and retroreflectivesheeting. The retroreflective sheeting may be derived from the laminaeor have the same optical features described with reference to a, lamina.Retroreflective sheeting may have cube corner elements, cube cornercavities, or combinations thereof.

[0024] Hence, in other embodiments, the invention disclosesretroreflective sheeting comprising a row of preferred geometry cubecorner elements having faces defined by grooves wherein adjacent sidegrooves range from being nominally parallel to nonparallel by less than1° and have included angles that differ by at least 2°. In otherembodiments, the retroreflective sheeting comprises a row of cube cornerelements wherein a first cube corner element is canted having analignment angle between 45° and 135° and a second adjacent cube iscanted having an alignment angles between 225° and 315°. In yet otherembodiments, the retroreflective sheeting comprises a row of preferredgeometry cube corner elements having faces defined by a side groove setwherein at least two grooves within the set are nonparallel by amountsranging from greater than nominally parallel to about 1°. In each ofthese embodiments, the sheeting preferably further comprises thefeatures described with reference to the lamina or laminae.

[0025] In another aspect, the invention discloses retroreflectivesheeting comprising a pair of adjacent rows of preferred geometry cubecorner elements wherein adjacent elements in a row have at least onedihedral edge that ranges from being nominally parallel to nonparallelby less than 1° and wherein the pair of rows comprise at least two typesof matched pairs.

[0026] In preferred embodiments, the retroreflective sheeting disclosedhas improved properties. In one embodiment, the retroreflective sheetingexhibits a uniformity index of at least 1. Such uniformity can beobtained without tiling in more than one orientation. The uniformityindex is preferably at least 3 and more preferably at least 5. In otherpreferred embodiments, the retroreflective sheeting comprises an arrayof preferred geometry cube corner elements that exhibits an averagebrightness at 0° and 90° orientation according to ASTM D4596-1a of atleast 375 candelas/lux/m² for an entrance angle of −4° and anobservation angle of 0.5°. Preferably, the sheeting exhibits improvedbrightness at other observation angles as well.

[0027] The invention further discloses any combination of featuresdescribed herein.

BRIEF DESCRIPTION OF THE DRAWINGS

[0028]FIG. 1 is a perspective view of an exemplary single lamina priorto formation of cube corner elements.

[0029]FIG. 2 is an end view of an exemplary single lamina following theformation of a first groove set.

[0030]FIG. 3 is a side view of an exemplary single lamina following theformation of a first groove set.

[0031]FIG. 4 is a top view of an exemplary single lamina following theformation of a first groove set and a second groove set.

[0032]FIG. 5 is a top view of an exemplary single lamina following theformation of a first groove set and primary groove face.

[0033]FIG. 6 is a top plan view of an exemplary assembly of four laminaecomprising a first groove set and a third primary groove wherein thecube corners have been canted sideways.

[0034]FIG. 7 is a side view depicting the symmetry axes of a pair ofadjacent sideways canted cubes on a lamina.

[0035]FIG. 8 is a perspective view of four laminae wherein the cubecorners have been canted sideways.

[0036]FIG. 9 is a perspective view of four laminae wherein the cubecorners have been canted sideways and the laminae have been assembled inopposing orientations.

[0037]FIG. 10a is a representation of a backward canted cube cornerelement.

[0038]FIG. 10b is a representation of a sideways canted cube cornerelement.

[0039]FIG. 10c is a representation of a forward canted cube cornerelement.

[0040]FIG. 11 depicts a top plan view of an assembly of laminae whereinthe cube corners have been canted forward in a plane normal to the edgeof the lamina.

[0041]FIG. 12 depicts a top plan view of an assembly of laminae whereinthe cube corners have been canted backward in a plane normal to the edgeof the lamina.

[0042]FIG. 13 depicts an isointensity plot showing the predicted lightreturn contours for a matched pair of cube corner elements comprised ofpolycarbonate that have been canted forward 9.74°.

[0043]FIG. 14 depicts an isointensity plot showing the predicted lightreturn contours for a matched pair of cube corner elements comprised ofpolycarbonate that have been canted backward 7.74°.

[0044]FIG. 15 depicts an isointensity plot showing the predicted lightreturn contours for two opposing laminae that comprise polycarbonatecubes that have been canted sideways 4.41°.

[0045]FIG. 16 depicts an isointensity plot showing the predicted lightreturn contours for two opposing laminae that comprise polycarbonatecubes that have been canted sideways 5.23°.

[0046]FIG. 17 depicts an isointensity plot showing the predicted lightreturn contours for two opposing laminae that comprises polycarbonatecubes that have been canted sideways 6.03°.

[0047]FIG. 18 depicts an isointensity plot showing the predicted lightreturn contours for two opposing laminae that comprise polycarbonatecubes that have been canted sideways 7.33°.

[0048]FIG. 19 depicts an isointensity plot showing the predicted lightreturn contours for an assembly of laminae that comprises polycarbonatecubes that have been canted sideways 9.74°.

[0049]FIG. 20 is a plot of alignment angle versus uniformity index.

[0050]FIG. 21 depicts a top plan view of a lamina having skewed sidegrooves.

[0051]FIG. 22 depicts each of the dihedral angles of a representativecube corner element.

[0052]FIG. 23 depicts a side view of a cube corner element of a laminadepicting positive and negative inclination.

[0053]FIG. 24 depicts a spot diagram for cubes that are backward cantedby 7.47 degrees with angle errors of the primary groove ranging from 2to 10 arc minutes.

[0054]FIG. 25 depicts a spot diagram for cubes that are backward cantedby 7.47 degrees with angle errors of the side grooves ranging from 0 to−20 arc minutes.

[0055]FIG. 26 depicts a spot diagram for cubes that are backward cantedby 7.47 degrees with a combination of primary groove and side grooveangle errors.

[0056]FIG. 27 depicts a spot diagram for cubes that are backward cantedby 7.47 degrees wherein the side grooves comprise a constant skew of 7arc minutes, a side groove angle error of +1.5 arc minutes andinclination varied in a repeating pattern over every four grooves.

[0057]FIG. 28 depicts a spot diagram for cubes of the same geometry asFIG. 29 except that the skew is −7 arc minutes rather than +7 arcminutes.

[0058]FIG. 29 depicts a spot diagram for the combination of FIG. 27 andFIG. 28.

[0059]FIG. 30 comprises the same angle errors, skews, and inclinationsas FIG. 29 except that the cubes are forward canted by 7.47 degrees.

[0060]FIG. 31 depicts a spot diagram for cubes that are sideways cantedby 6.02 degrees having various skews and inclinations.

[0061] The drawings, particularly of the lamina(e), are illustrative andthus not necessary representative of actual size. For example thedrawing(s) may be an enlarged lamina or enlarged portion of a lamina.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0062] The present invention relates to a lamina and laminae comprisingcube corner elements, a tool comprising an assembly of laminae andreplicas. There invention further relates to retroreflective sheeting.

[0063] The retroreflective sheeting is preferably prepared from a mastermold manufactured with a technique that employs laminae. Accordingly, atleast a portion and preferably substantially all the cube cornerelements of the lamina(e) and retroreflective sheeting are full cubesthat are not truncated. In one aspect, the base of full cube elements inplan view are not triangular. In another aspect, the non-dihedral edgesof full cube elements are characteristically not all in the same plane(i.e. not coplanar). Such cube corner elements are preferably “preferredgeometry (PG) cube corner elements”.

[0064] A PG cube corner element may be defined in the context of astructured surface of cube corner elements that extends along areference plane. For the purposes of this application, a PG cube cornerelement means a cube corner element that has at least one non-dihedraledge that: (1) is nonparallel to the reference plane; and (2) issubstantially parallel to an adjacent non-dihedral edge of a neighboringcube corner element. A cube corner element whose three reflective facescomprise rectangles (inclusive of squares), trapezoids or pentagons areexamples of PG cube corner elements. “Reference plane” with respect tothe definition of a PG cube corner element refers to a plane or othersurface that approximates a plane in the vicinity of a group of adjacentcube corner elements or other geometric structures, the cube cornerelements or geometric structures being disposed along the plane. In thecase of a single lamina, the group of adjacent cube corner elementsconsists of a single row or pair of rows. In the case of assembledlaminae, the group of adjacent cube corner elements includes the cubecorner elements of a single lamina and the adjacent contacting laminae.In the case of sheeting, the group of adjacent cube corner elementsgenerally covers an area that is discernible to the human eye (e.g.preferably at least 1 mm²) and preferably the entire dimensions of thesheeting.

[0065] “Entrance angle” refers to the angle between the reference axis(i.e. the normal vector to the retroreflective sample) and the axis ofthe incident light.

[0066] “Orientation” refers to the angle through which the sample may berotated about the reference axis from the initial zero degreeorientation of a datum mark.

[0067] Lamina(e) refers to at least two lamina. “Lamina” refers to athin plate having length and height at least about 10 times itsthickness (preferably at least 100, 200, 300, 400, 500 times itsthickness). The invention is not limited to any particular dimensions oflamina(e). In the case of lamina intended for use in the manufacture ofretroreflective sheeting, optimal dimensions may be constrained by theoptical requirements of the final design (e.g. cube corner structures).In general the lamina has a thickness of less than 0.25 inches (6.35 mm)and preferably less than 0.125 inches (3.175 mm). The thickness of thelamina is preferably less than about 0.020 inches (0.508 mm) and morepreferably less than about 0.010 inches (0.254 mm). Typically, thethickness of the lamina is at least about 0.001 inches (0.0254 mm) andmore preferably at least about 0.003 inches (0.0762 mm). The laminaranges in length from about 1 inch (25.4 mm) to about 20 inches (50.8cm) and is typically less than 6 inches (15.24 cm). The height of thelamina typically ranges from about 0.5 inches (12.7 mm) to about 3inches (7.62 cm) and is more typically less than about 2 inches (5.08cm).

[0068] With reference to FIGS. 1-8 lamina 10 includes a first majorsurface 12 and an opposing second major surface 14. Lamina 10 furtherincludes a working surface 16 and an opposing bottom surface 18extending between first major surface 12 and second major surface 14.Lamina 10 further includes a first end surface 20 and an opposing secondend surface 22. In a preferred embodiment, lamina 10 is a rightrectangular polyhedron wherein opposing surfaces are substantiallyparallel. However, it will be appreciated that opposing surfaces oflamina 10 need not be parallel.

[0069] Lamina 10 can be characterized in three-dimensional space bysuperimposing a Cartesian coordinate system onto its structure. A firstreference plane 24 is centered between major surfaces 12 and 14. Firstreference plane 24, referred to as the x-z plane, has the y-axis as itsnormal vector. A second reference plane 26, referred to as the x-yplane, extends substantially coplanar with working surface 16 of lamina10 and has the z-axis as its normal vector. A third reference plane 28,referred to as the y-z plane, is centered between first end surface 20and second end surface 22 and has the x-axis as its normal vector. Forthe sake of clarity, various geometric attributes of the presentinvention will be described with reference to the Cartesian referenceplanes as set forth herein. However, it will be appreciated that suchgeometric attributes can be described using other coordinate systems orwith reference to the structure of the lamina.

[0070] The lamina(e) of the present invention preferably comprise cubecorner elements having faces formed from, and thus comprise ,a firstgroove set, an optional second groove set, and preferably a thirdprimary groove (e.g. primary groove face).

[0071]FIGS. 2-9 illustrate a structured surface comprising a pluralityof cube corner elements in the working surface 16 of lamina 10. Ingeneral, a first groove set comprising at least two and preferably aplurality of grooves 30 a, 30 b, 30 c, etc. (collectively referred to as30) are formed in working surface 16 of lamina 10. The grooves 30 areformed such that the respective groove vertices 33 and the respectivefirst reference edges 36 extend along an axis that intersects the firstmajor surface 12 and the working surface 16 of lamina 10. Althoughworking surface 16 of the lamina 10 may include a portion that remainsunaltered (i.e. unstructured), it is preferred that working surface 16is substantially free of unstructured surface portions.

[0072] The direction of a particular groove is defined by a vectoraligned with the groove vertex. The groove direction vector may bedefined by its components in the x, y and z directions, the x-axis beingperpendicular to reference plane 28 and the y-axis being perpendicularto reference plane 24. For example, the groove direction for groove 30 bis defined by a vector aligned with groove vertex 33 b. It is importantto note that groove vertices may appear parallel to each other in topplan view even though the grooves are not parallel (i.e. differentz-direction component).

[0073] As used herein, the term “groove set” refers to grooves formed inworking surface 16 of the lamina 10 that range from being nominallyparallel to non-parallel to within 10 to the adjacent grooves in thegroove set. As used herein “adjacent groove” refers to the closestgroove that is nominally parallel or non-parallel to within 1°.Alternatively or in addition thereto, the grooves of a groove set mayrange from being nominally parallel to non-parallel to within 1° toparticular reference planes as will subsequently be described.Accordingly, each characteristic with regard to an individual grooveand/or the grooves of a groove set (e.g. perpendicular, angle, etc.)will be understood to have this same degree of potential deviation.Nominally parallel grooves are grooves wherein no purposeful variationhas been introduced within the degree of precision of the groove-formingmachine. The grooves of the groove set may also comprise smallpurposeful variations for the purpose of introducing multiplenon-orthogonality (MNO) such as included angle errors, and/or skew,and/or inclination as will subsequently be described in greater detail.

[0074] Referring to FIGS. 3-9, the first groove set comprises grooves 30a, 30 b, 30 c, etc. (collectively referred to by the reference numeral30) that define first groove surfaces 32 a, 32 b, 32 c, etc.(collectively referred to as 32) and second groove surfaces 34 b, 34 c,34 d, etc. (collectively referred to as 34) that intersect at groovevertices 33 b, 33 c, 33 d, etc. (collectively referred to as 33). At theedge of the lamina, the groove forming operation may form a singlegroove surface 32 a.

[0075] In another embodiment depicted in FIG. 4, lamina 10 mayoptionally further comprise a second groove set comprising at least twoand preferably a plurality of adjacent grooves, collectively referred toas 38) also formed in the working surface 16 of lamina 10. In thisembodiment, the first and second groove sets intersect approximatelyalong a first reference plane 24 to form a structured surface includinga plurality of alternating peaks and v-shaped valleys. Alternatively,the peaks and v-shaped valleys can be off-set with respect to eachother. Grooves 38 define third groove surfaces 40 a, 40 b, etc.(collectively referred to as 40) and fourth groove surfaces 42 b, 42 c,etc. (collectively referred to as 42) that intersect at groove vertices41 b, 41 c, etc. (collectively referred to as 41) as shown. At the edgeof the lamina, the groove forming operation may form a single groovesurface 40 a.

[0076] Both these first and second groove sets may also be referred toherein as “side grooves”. As used herein side grooves refer to a grooveset wherein the groove(s) range from being nominally parallel tonon-parallel to within 1°, per their respective direction vectors, tothe adjacent side grooves of the side groove set. Alternatively or inaddition thereto, side grooves refers to a groove that range from beingnominally parallel to reference plane 28 to nonparallel to referenceplane 28 to within 1°. Side grooves are typically perpendicular toreference plane 24 to this same degree of deviation in plan view.Depending on whether the side grooves are nominally parallel ornon-parallel within 1°, individual elements in the replicated assembledmaster typically have the shape of trapezoids, rectangles,parallelograms and pentagons, and hexagons when viewed in plan view witha microscope or by measuring the dihedral angles or parallelism of theside grooves with an interferometer. Suitable interferometers willsubsequently be described.

[0077] Although the third face of the elements may comprise workingsurface 12 or 14, such as describe in EP 0 844 056 A1 (Mimura et al.),the lamina preferably comprises a primary groove face 50 that extendssubstantially the full length of the lamina. Regardless of whether thethird face is a working surface (i.e. 12 or 14) of the lamina or aprimary groove face, the third face of each element within a rowpreferably share a common plane. With reference to FIG. 5-6 and 8-9,primary groove face 50 ranges from being nominally perpendicular tofaces 32, 34, 40 and 42 to non-perpendicular to within 1°. Formation ofprimary groove face 50 results in a structured surface that includes aplurality of cube corner elements having three perpendicular orapproximately perpendicular optical faces on the lamina. A single laminamay have a single primary groove face, a pair of groove faces onopposing sides and/or a primary groove along the intersection of workingsurface 16 with reference plane 24 that concurrently provides a pair ofprimary groove faces (e.g. FIG. 4). The primary groove is preferablyparallel to reference plane 26 to within 1°.

[0078] A pair of single laminae with opposing orientations andpreferably multiple laminae with opposing orientations are typicallyassembled into a master tool such that their respective primary groovefaces form a primary groove. For example, as depicted in FIGS. 6 and8-9, four laminae (i.e. laminae 100, 200, 300 and 400 are preferablyassembled such that every other pair of laminae are positioned inopposing orientations (i.e. the cube corner elements of lamina 100 arein opposing orientation with the cube corner elements of lamina 200 andthe cube corner elements of lamina 300 are in opposing orientation withthe cube corner elements of lamina 400). Further, the pairs of laminaehaving opposing orientation are positioned such that their respectiveprimary groove faces 50 form primary groove 52. Preferably the opposinglaminae are positioned in a configuration (e.g. 34 b aligned with 42 b)in order to minimize the formation of vertical walls.

[0079] After formation of the groove sets, working surface 16 ismicrostructured. As used herein, “microstructured” refers to at leastone major surface of the sheeting comprising structures having a lateraldimension (e.g. distance between groove vertices of the cube cornerstructures) of less than 0.25 inches (6.35 mm), preferably less than0.125 inches (3.175 mm) and more preferably less than 0.04 inches (1mm). The lateral dimension of cube corner elements, is preferably lessthan 0.020 inches (0.508 mm) and more preferably less than 0.007 inches(0.1778 mm). Accordingly, the respective groove vertices 33 and 41 arepreferably separated by this same distance throughout the groove otherthan the small variations resulting from non-parallel grooves. Themicrostructures have an average height ranging from about 0.001 inches(0.0254 mm) to 0.010 inches (0.254 mm), with a height of less than 0.004inches (0.1016 mm) being most typical. Further, the lateral dimension ofa cube corner microstructure is typically at least 0.0005 inches (0.0127mm). Cube corner microstructures may comprise either cube cornercavities or, preferably, cube corner elements having peaks.

[0080] In one embodiment, as depicted in FIG. 3-9, the present inventionrelates to a lamina and laminae comprising a side groove set whereinadjacent grooves comprise different included angles. “Included angle”refers to the angle formed between the two faces of a V-shaped groovethat intersect at the groove vertex. The included angle is typically afunction of the geometry of the diamond-cutting tool and its positionrelative to the direction of cut. Hence, a different diamond tool istypically employed for each different included angle. Alternatively, yetmore time consuming, different included angles may be formed by makingmultiple cuts within the same groove. The included angles of a firstgroove (e.g. side groove) differs from an adjacent groove (e.g. secondside groove) by at least 2° (e.g. 3°, 4°, 5°, 6°, 7°, 8°, 9°) preferablyat least 10° (e.g. 11°, 12°, 13°, 14°), and more preferably by at least15° (e.g. 16°, 17°, 18°, 19°, 20°). Accordingly, the difference inincluded angle is substantially larger than differences in includedangles that would arise from purposeful angle errors introduced for thepurpose of non-orthogonality. Further, the difference in included anglesis typically less than 70° (e.g. 65°, 60°, 50°), preferably less than55°, more preferably less than 50°, and even more preferably less than40°.

[0081] In one aspect, the differing included angles (e.g. of adjacentside grooves) are arranged in a repeating pattern to minimize the numberof different diamond cutting tools needed. In such embodiment, the sumof adjacent side groove angles is about 180°. In a preferred embodiment,the lamina comprises a first sub-set of side grooves having an includedangle greater than 90° alternated with second sub-set of side grooveshaving an included angle less than 90°. In doing so, the included angleof a first groove is typically greater than 90° by an amount of at leastabout 5°, and preferably by an amount ranging from about 10° to about20°; whereas the included angle of the adjacent groove is less than 90°by about the same amount.

[0082] Although, the lamina may further comprise more than two sub-setsand/or side grooves having included angles of nominally 90°, the laminais preferably substantially free of side grooves having an includedangle of nominally 90°. In a preferred embodiment, the lamina comprisesan alternating pair of side grooves (e.g. 75.226° and 104.774°) andthus, only necessitates the use of two different diamonds to form thetotality of side grooves. Accordingly, with reference to FIGS. 6-9,every other side grooves, i.e. 30 a, 30 c, 30 e, etc. has an includedangle of 75.226° alternated with the remaining side grooves, i.e. 30 b,30 d, etc. having an included angle of 104.774°. As will subsequently bedescribed in further detail, employing differing included angles in thismanner improves the uniformity index.

[0083] In another aspect, alternatively or in combination with thediffering included angles (e.g. of adjacent side grooves) being arrangedin a repeating pattern, the resulting cube corner elements have facesthat intersect at a common peak height, meaning that cube peaks (e.g.36) are within the same plane to within 3-4 microns. It is surmised thata common peak height contribute to improved durability when handling thetooling or sheeting by evenly distributing the load.

[0084] Alternatively or in combination thereof, the lamina comprisessideways canted cube corner elements. For cube corner elements that aresolely canted forward or backward, the symmetry axes are canted ortilted in a cant plane parallel with reference plane 28. The cant planefor a cube corner element is the plane that is both normal to referenceplane 26 and contains the symmetry axis of the cube. Accordingly, thenormal vector defining the cant plane has a y component of substantiallyzero for cube corner elements that are solely canted forward orbackward. In the case of cube corner elements that are solely cantedsideways, the symmetry axes of the cubes are canted in a plane that issubstantially parallel to reference plane 24 and thus, the normal vectordefining the cant plane has an x component of substantially zero.

[0085] The projection of the symmetry axis in the x-y plane mayalternatively be used to characterize the direction of cant. Thesymmetry axis is defined as the vector that trisects the three cubecorner faces forming an equal angle with each of these three faces.FIGS. 10a-10 c depict three different cube corner geometries in planview that are solely backward canted, solely sideways canted, and solelyforward canted, respectively. In these figures the cube peak extends outof the page and the projection of the symmetry axis (extending into thepage from the cube peak) in the x-y plane is shown by the arrow. Thealignment angle is measured counterclockwise in this view from thedihedral edge 11 (e.g. dihedral 2-3) that is approximately normal to aside of the cube in plan view. In the case of backward canting in theabsence of sideways canting, the alignment angle is 0 degrees, whereasin the case of forward canting in the absence of sideways canting thealignment angle is 180 degrees. For a cube that has been canted sidewaysin the absence of forward or backward canting, the alignment angle iseither 90° (as shown in FIG. 10b) or 270°. Alignment angle is 90° whenthe projection of the symmetry axis points to the right (FIG. 10b) and270° when the projection of the symmetry axis points to the left.

[0086] Alternatively, the cube may be canted such that the cant planenormal vector comprises both an x-component and y-component (i.e.x-component and y-component are each not equal to zero). At an alignmentangle between 0° and 45° or between 0° and 315° the backward cantcomponent is predominant with the backward cant component and sidewayscant component being equal at an alignment angle of 45° or 315°. Furtherat an alignment angle between 135° and 225°, the forward cant componentis predominant with the forward cant component and sideways cantcomponent being equal at 135° and at 225°. Accordingly, cant planescomprising a predominant sideways cant component have an alignment anglebetween 45° and 135° or between 225° and 315°. Hence, a cube cornerelement is predominantly sideways canting when the absolute value of they-component of the cant plane normal vector is greater than the absolutevalue of the x-component of the cant plane normal vector.

[0087] For embodiments wherein the sideways canted cubes are formed froman alternating pair of side grooves having different included anglecubes where the cant plane is parallel to reference plane 24 theadjacent cubes within a given lamina (e.g. α-β or α′-β′) are canted inthe same or parallel planes. However, in general, if there is an xcomponent to the cant plane normal vector, then adjacent cubes within aparticular lamina are not canted in the same plane. Rather, the cubecorner matched pairs are canted in the same or parallel planes (i.e.α-α′ or β-β′). Preferably, the cube corner elements of any given laminahave only two different alignment angles, e.g. derived from adjacentside grooves comprising different included angles. The alignment anglefor the sideways canting example in FIG. 10b is 90°, corresponding tothe β-β′ cubes in FIG. 6. Similarly, the alignment angle for the α-α′sideways canted cubes in FIG. 6 is 270° (not shown).

[0088]FIG. 11 depicts laminae wherein the cubes are canted forward;whereas FIG. 12 depicts laminae wherein the cubes are canted backward.Cube designs canted in this manner result in a single type of matchedopposing cube pairs. The cube 54 a of FIG. 11 with faces 64 a and 62 bis the same as the cube 54 b with faces 64 b and 62 c that is the sameas cube 54 c with faces 64 c and 62 d, etc. Accordingly, all the cubeswithin the same row are the same, the row being parallel to referenceplane 24. The cube 54 a is a matched opposing pair with cube 56 a havingfaces 66 e and 68 d. Further, the cube 54 b is a matched opposing pairwith cube 56 b. Likewise, cube 54 c is a matched opposing pair with cube56 c. Similarly, cube 57 of FIG. 12 is a matched opposing pair with cube58. Matched pairs result when 180° rotation of a first cube about anaxis normal to the plane of the structured surface will result in a cubethat is super-imposable onto a second cube. Matched pairs need notnecessarily be directly adjacent or contacting within the group of cubecorner elements. Matched pairs typically provide retroreflectiveperformance that is symmetric with respect to positive or negativechanges in entrance angle for opposing orientations.

[0089] In contrast, sideways canting results in a cube design comprisingtwo different cube orientations within the same row and thus created bythe same side groove set. For a single lamina comprising both the firstand second set of side grooves or a pair of adjacent laminae assembledin opposing orientations, the laminae comprise four distinctly differentcubes and two different matched pairs, as depicted in FIGS. 6, 8-9. Thefour distinctly different cubes are identified as cubes alpha (α) havingfaces 32 b and 34 c, beta (β) having faces 32 c and 34 d, alpha prime(α′) having faces 40 c and 42 d, and beta prime (β′) having faces 40 band 42 c. Cubes (α, α′) are a matched pair with the same geometry whenrotated 180° such that the cube faces are parallel, as are cubes (β,β′). Further, the cubes on adjacent laminae (e.g. 100, 200) areconfigured in opposing orientations. Although the symmetry axis of thecubes is tipped sideways, the bisector plane of the side grooves (i.e.the plane that divides the groove into two equal parts) preferablyranges from being nominally parallel to the bisector plane of anadjacent side groove (i.e. mutually parallel) to being nonparallelwithin 1°. Further, each bisector plane ranges from being nominallyparallel to reference plane 28 to being nonparallel to reference plane28 within 1°.

[0090]FIGS. 13-14 are isobrightness contour graphs illustrating thepredicted total light return for a retroreflective cube corner elementmatched pair formed from a material having an index of refraction of1.59 at varying entrance angles and orientation angles. In FIG. 13 thematched pair is forward canted 9.74° (e.g. cube corner elements 54, 56of FIG. 11). In FIG. 14, the matched pair is backward canted 7.47° (e.g.cube corner elements 57, 58 of FIG. 12). FIGS. 15-19 are isobrightnesscontour graph illustrating the predicted total light return for laminaecomprising retroreflective cube corner elements formed from a materialhaving an index of refraction of 1.59 at varying entrance angles andorientation angles where the cube corner elements are canted sideways4.41°, 5.23°, 6.03°, 7.33°, and 9.74°, respectively for alignment anglesof 90° and 270°. An alternating pair of side grooves (i.e. 75.226° and104.774°) is utilized for FIG. 17 to produce cube corner elements thatare sideways canted by 6.03°. The sideways canted arrays have two typesof matched pairs, the β-β′ and α-α′ as depicted in FIG. 6. These matchedpairs have alignment angles of 90° and 270° respectively. In each ofFIGS. 15-19, the isobrightness contour graph is for laminae having thesame (i.e. vertical) orientation as depicted in FIGS. 6, 11 and 12.

[0091] Predicted total light return for a cube corner matched pair arraymay be calculated from a knowledge of percent active area and rayintensity. Total light return is defined as the product of percentactive area and ray intensity. Total light return for directly machinedcube corner arrays is described by Stamm U.S. Pat. No. 3,812,706.

[0092] For an initial unitary light ray intensity, losses may resultfrom two pass transmissions through the front surface of the sheetingand from reflection losses at each of the three cube surfaces. Frontsurface transmission losses for near normal incidence-and a sheetingrefractive index of about 1.59 are roughly 0.10 (roughly 0.90transmission). Reflection losses for cubes that have been reflectivelycoated depend for example on the type of coating and the angle ofincidence relative to the cube surface normal. Typical reflectioncoefficients for aluminum reflectively coated cube surfaces are roughly0.85 to 0.9 at each of the cube surfaces. Reflection losses for cubesthat rely on total internal reflection are essentially zero (essentially100% reflection). However, if the angle of incidence of a light rayrelative to the cube surface normal is less than the critical angle,then total internal reflection can break down and a significant amountof light may pass through the cube surface. Critical angle is a functionof the refractive index of the cube material and of the index of thematerial behind the cube (typically air). Standard optics texts such asHecht, “Optics”, 2nd edition, Addison Wesley, 1987 explain front surfacetransmission losses and total internal reflection. Effective area for asingle or individual cube corner element may be determined by, and isequal to, the topological intersection of the projection of the threecube corner surfaces on a plane normal to the refracted incident raywith the projection of the image surfaces of the third reflection on thesame plane. One procedure for determining effective aperture isdiscussed for example by Eckhardt, Applied Optics, v. 10, n. 7, July1971, pg. 1559-1566. Straubel U.S. Pat. No. 835,648 also discusses theconcept of effective area or aperture. Percent active area for a singlecube corner element is then defined as the effective area divided by thetotal area of the projection of the cube corner surfaces. Percent activearea may be calculated using optical modeling techniques known to thoseof ordinary skill in the optical arts or may be determined numericallyusing conventional ray tracing techniques. Percent active area for acube corner matched pair array may be calculated by averaging thepercent active area of the two individual cube corner elements in thematched pair. Alternatively stated, percent active aperture equals thearea of a cube corner array that is retroreflecting light divided by thetotal area of the array. Percent active area is affected for example bycube geometry, refractive index, angle of incidence, and sheetingorientation.

[0093] Referring to FIG. 13 vector V₁ represents the plane that isnormal to reference plane 26 and includes the symmetry axes of cubecorner elements 54, 56 in FIG. 11. For example, V₁ lies in a planesubstantially normal to the primary groove vertex 51 formed by theintersection of the primary groove faces 50. The concentricisobrightness curves represent the predicted total light return of thearray of cube corner elements 54, 56 at various combinations of entranceangles and orientation angles. Radial movement from the center of theplot represents increasing entrance angles, while circumferentialmovement represents changing the orientation of the cube corner elementwith respect to the light source. The innermost isobrightness curvedemarcates the set of entrance angles at which a matched pair of cubecorner elements 54, 56 exhibit 70% total light return. Successivelyoutlying isobrightness curves demarcate entrance and orientation angleswith successively lower percentages.

[0094] A single matched pair of forward or backward canted cubestypically have two planes (i.e. V₁ and V₂) of broad entrance angularitythat are substantially perpendicular to one another. Forward cantingresults in the principle planes of entrance angularity being horizontaland vertical as shown in FIG. 13. The amount of light returned at higherentrance angles varies considerably with orientation and is particularlylow between the planes of best entrance angularity. Similarly, backwardcanting results in the principle planes of entrance angularity (i.e. V₃and V₄) oriented at roughly 45° to the edge of the lamina as shown inFIG. 14. Similarly, the amount of light returned at higher entranceangles varies considerably with orientation and is particularly lowbetween the planes of best entrance angularity.

[0095]FIGS. 15-19 depict the predicted total light return (TLR)isointensity contours for a pair of opposing laminae having sidewayscanted cubes. The light return is more uniform as indicated by the shapeof the plot approaching circular, in comparison to the isointensityplots of forward and backward canted cubes of FIGS. 13 and 14. FIGS.15-19 depict substantially higher light return at the locations of lowlight return evident in FIGS. 13 and 14. Accordingly, good retention ofTLR at higher entrance angles (up to at least 45° entrance) ispredicted. This improved orientation performance is desirable forsigning products since the signs are commonly positioned at orientationsof 0°, 45° and 90°. Prior to the present invention, a common method forimproving the uniformity of total light return with respect toorientation has been tiling, i.e. placing a multiplicity of smalltooling sections in more than one orientation, such as described forexample in U.S. Pat. No. 5,936,770. Sideways canted cube corner arrayscan improve the uniformity of total light return, without the need fortiling and thus the array is substantially free of tiling in more thanone orientation.

[0096] In order to compare the uniformity of total light return (TLR) ofvarious optical designs, the average TLR at orientations of 0°, 45° and90° may be divided by the range of TLR at orientations of 0°, 45° and90°, i.e. the difference between the maximum and minimum TLR at theseangles, all at a fixed entrance angle. The entrance angle is preferablyat least 30° or greater, and more preferably 40° or greater. Preferreddesigns exhibit the maximum ratio of average TLR relative to TLR range.This ratio, i.e. “uniformity index (UI)” was calculated for a 40°entrance angle for the forward and backward canted cubes of FIGS. 13 and14, respectively as well as for the sideways canted cubes having variousdegrees of tilt of FIGS. 15-19. For Table 1 the spacing of the sidegrooves is equal to the thickness of the lamina (i.e. aspect ratio=1).The calculated uniformity index is summarized in Table 1 as follows:TABLE 1 Forward Backward Sideways (alignment angle = 90°) Amount of cant9.74 7.47 4.41 5.23 6.03 7.33 9.74 (arc minutes) Avg. TLR 0.210 0.1330.160 0.184 0.209 0.180 0.166 (0/45/90) TLR Range 0.294 0.154 0.0900.023 0.034 0.167 0.190 (0/45/90) UI 0.71 0.87 1.79 8.02 6.23 1.08 0.88${{Uniformity}\quad {Index}\quad ({UI})} = \frac{{{Average}\quad {TLR}\quad {of}\quad 0{^\circ}},{45{^\circ}},{90{^\circ}}}{{{Range}\quad {at}\quad 0{^\circ}},{45{^\circ}\quad {and}\quad 90{^\circ}}}$

[0097] Improved orientation uniformity results when the uniformity indexis greater than 1. Preferably, the uniformity index is greater than 3(e.g. 4), and more preferably greater than 5 (e.g. 6, 7, 8). Uniformityindex will vary as a function of variables such as cube geometry (e.g.amount and type of cant, type of cube, cube shape in plan view, locationof cube peak within aperture, cube dimensions), entrance angle, andrefractive index.

[0098]FIG. 20 illustrates the uniformity index plotted versus alignmentangle for canted cube corner arrays with varying amounts of cant andvarying x and y components for their cant plane normal vectors. Thearrays have two types of matched pairs, similar to the β-β′ and α-α′ asdepicted in FIG. 6, although these cubes and/or pairs may not bemutually adjacent. The cubes in each matched pair have substantially thesame alignment angle. Alignment angles for the two types of matchedpairs differ from 0° or 180° by the same amount. For example, for analignment angle of 100° (differing from 180° by 80°) for a first cube orfirst matched pair the second (e.g. adjacent) cube or second matchedpair would have an alignment angle of 260° (also differing from 180° by80°).

[0099]FIG. 20 shows that the average TLR for polycarbonate (having anindex of refraction of 1.59) as well as the uniformity index aremaximized for cube geometries having a predominant sideways cantingcomponent, i.e. the range roughly centered about alignment angles of 90°and 270°. Note that alignment angles between 0° and 180° are presentedon the X or horizontal axis of FIG. 20 from left to right. Alignmentangles increasing from 180° to 360° degrees are plotted from right toleft.

[0100] Preferably, the alignment angle is greater than 50° (e.g. 51°,52°, 53°, 54°), more preferably greater than 55° (e.g. 56°, 57°, 58°,59°), and even more preferably greater than 60°. Further the alignmentangle is preferably less than 130° (e.g. 129°, 128°, 127°, 126°) andmore preferably less than 125° (e.g. 124°, 123°, 122°, 121°), and evenmore preferably less than 120°. Likewise the alignment angle ispreferably greater than 230° (e.g. 231°, 232°, 233°, 234°), and morepreferably greater than 235° (e.g. 236°, 237°, 238°, 239°), and evenmore preferably greater than 240° . Further the alignment angle ispreferably less than 310° (e.g. 309°, 308°, 307°, 306°) and morepreferably less than 305° (e.g. 304°, 303°, 302°, 301°) and even morepreferably less than 300°.

[0101] The amount of tilt of the cube symmetry axes relative to a vectorperpendicular to the plane of the cubes is at least 2° and preferablygreater than 3°. Further, the amount of tilt is preferably less than90°. Accordingly, the most preferred amount of tilt ranges from about3.5° to about 8.5° including any interval having end points selectedfrom 3.6°, 3.7°, 3.8°, 3.9°, 4.0°, 4.1°, 4.2°, 4.3°, 4.4° and 4.5°combined with end points selected from 7.5°, 7.6°, 7.7°, 7.8°, 7.9°,8.0°, 8.1°, 8.2°, 8.30° and 8.4°. Cube geometries that may be employedto produce these differing amounts of sideways cant are summarized inTable 2. The alignment angle may be 90° or 270° for each amount of cant.TABLE 2 Amount of Side groove Side groove Side groove Side groove CantSub-set 1 Sub-set 2 Sub-set 1 Sub-set 2 (°) Half angle (°) 1/2 angle (°)Full angle (°) Full angle (°) 4.41 39.591 50.409 79.182 100.818 5.2338.591 51.409 77.182 102.818 6.03 37.613 52.387 75.226 104.774 7.3336.009 53.991 72.018 107.982 9.74 33.046 56.954 66.092 113.908

[0102] Although differing included angles alone or in combination withthe previously described sideways canting provide improved brightnessuniformity in TLR with respect to changes in orientation angle over arange of entrance angles, it is also preferred to improve theobservation angularity or divergence profile of the sheeting. Thisinvolves improving the spread of the retroreflected light relative tothe source (typically, vehicle headlights). As previously describedretroreflected light from cube corners spreads due to effects such asdiffraction (controlled by cube size), polarization (important in cubeswhich have not been coated with a specular reflector), andnon-orthogonality (deviation of the cube corner dihedral angles from 90°by amounts less than 1°). Spread of light due to non-orthogonality isparticularly important in (e.g. PG) cubes produced using laminae sincerelatively thin laminae would be required to fabricate cubes where thespreading of the return light was dominated by diffraction. Such thinlaminae are particularly difficult to handle during fabrication.

[0103] Alternatively, or in addition to the features previouslydescribed, in another embodiment the present invention relates to anindividual lamina, a master tool comprising the assembled laminae, aswell as replicas thereof including retroreflective replicas, comprisingside grooves wherein the side grooves comprise “skew” and/or“inclination”. Skew and/or inclination provides cubes with a variety ofcontrolled dihedral angle errors or multiple non-orthogonality (MNO) andthus improves the divergence profile of the finished product. As usedherein “skew” refers to the deviation from parallel with reference toreference plane 28.

[0104]FIG. 21 shows an exaggerated example in plan view of a singlelamina with one row of cube corner elements comprising skewed grooves.Side grooves 80 a and 80 b are cut with positive skew, grooves 80 c and80 e without skew, and groove 80 d with negative skew. The path of theside grooves 80 has been extended in FIG. 21 for clarity. Provided sidegrooves 80 a, 80 c, and 80 e have the same included angle (e.g. 75°,first groove sub-set), the same cutting tool or diamond can be used toform all of these grooves. Further, if the alternating grooves, namely80 b and 80 d have the same included angle (e.g. 105°, second groovesub-set) 80 b and 80 d can be cut with a second diamond. The primarygroove face 50 may also be cut with one of these diamonds if the primarygroove half angle is sufficiently close to the side groove half anglefor the first or second sub-sets. Optionally, one of the cutting toolsmay be rotated during cutting of the primary groove face in order toachieve the correct primary groove half angle. The primary groove faceis preferably aligned with the side of the lamina. Thus, the entirelamina can be cut incorporating MNO with the use of only two diamonds.Within each groove set skew can be easily varied during machining toproduce a range of dihedral angles. Cube corners in general have threedihedral angles attributed to the intersections of the three cube faces.The deviation of these angles from 90° is commonly termed the dihedralangle error and may be designated by dihedral 1-2, dihedral 1-3, anddihedral 2-3. In one naming convention, as depicted in FIG. 22, cubedihedral angle 1-3 is formed by the intersection of groove surface 82with primary groove face 50, cube dihedral 1-2 is formed by theintersection of groove surface 84 with primary groove face 50, and cubedihedral 2-3 is formed by the intersection of groove surface 84 withgroove surface 82. Alternatively, the same naming convention may beemployed wherein the third face is working surface 12 or 14 rather thana primary groove face. For a given groove, positive skew (80 a, 80 b)results in decreasing dihedral 1-3 and increasing dihedral 1-2 whilenegative skew results in increasing dihedral 1-3 and decreasing dihedral1-2.

[0105] For example, with reference to FIG. 21 four different cubes areformed. The first cube 86 a has groove surfaces (i.e. faces) 82 a and 84b, the second cube 86 b groove surfaces 82 b and 84 c, the third cube 86c groove surfaces 82 c and 84 d, and the fourth cube 86 d has groovesurfaces 82 d and 84 e. The intersection of groove surfaces 82 a, 82 b,and 84 d with groove face 50 are less than 90°, whereas the intersectionof groove surfaces 84 b and 82 d with groove face 50 are greater than90°. The intersection of groove surfaces 82 c, 84 c, and 84 e withgroove face 50 are 90° since grooves 80 c and 80 e are without skew. Thepreceding discussion assumes that the inclination (as will subsequentlybe defined) is the same for all the side grooves in FIG. 21 and equalszero. The (e.g. PG) cube corner elements are trapezoids orparallelograms in plan view shape as a result of using skewed groovesduring machining.

[0106] Alternatively, or in addition to the features previouslydescribed, the side grooves may comprise positive or negativeinclination. “Inclination” refers to the deviation in slope in referenceplane 28 of a particular side groove from the nominal orthogonal slope(i.e. the slope of the vector normal to the primary groove surface). Thedirection of a side groove is defined by a vector aligned with thevertex of said groove. Orthogonal slope is defined as the slope in whichthe vertex 90 of a groove would be parallel to the normal vector ofgroove face 50 (normal to groove face 50) for skew equal to zero. In onepossible naming convention, positive inclination 92 results indecreasing both dihedral 1-3 and dihedral 1-2 for a given side groovewhile negative inclination 94 results in increasing both dihedral 1-3and dihedral 1-2.

[0107] Combining skew and/or inclination during machining providessignificant flexibility in varying the dihedral angle errors of the cubecorner elements on a given lamina. Such flexibility is independent ofcant. Accordingly skew and/or inclination may be employed with uncantedcubes, forward canted cubes, backward canted cubes, as well as sidewayscanted cubes. The use of skew and/or inclination provides a distinctadvantage as it can be introduced during the machining of individuallamina without changing the tool (e.g. diamond) used to cut the sidegrooves. This can significantly reduce machining time as it typicallycan take hours to correctly set angles after a tool change. Furthermore,dihedral 1-2 and dihedral 1-3 may be varied in opposition using skewand/or inclination. “Varied in opposition” as used herein is defined asintentionally providing within a given cube corner on a lamina dihedral1-2 and 1-3 errors (differences from 90°) that differ in magnitudeand/or sign. The difference in magnitude is typically at least ¼ arcminutes, more preferably at least ½ arc minutes, and most preferably atleast 1 arc minutes. Hence the grooves are nonparallel by amount greaterthan nominally parallel. Further, the skew and/or inclination is suchthat the magnitude is no more than about 1° (i.e. 60 arc minutes).Further, the (e.g. side) grooves may comprise a variety of differentcomponents of skew and/or inclination along a single lamina.

[0108] Dihedral angle errors may also be varied by changing the halfangles of the primary or side grooves during machining. Half angle forside grooves is defined as the acute angle formed by the groove face anda plane normal to reference plane 26 that contains the groove vertex.Half angle for primary grooves or groove faces is defined as the acuteangle formed by the groove face and reference plane 24. Changing thehalf angle for the primary groove results in a change in slope of grooveface 50 via rotation about the x-axis. Changing the half angle for aside groove may be accomplished via either changing the included angleof the groove (the angle formed by opposing groove faces e.g. 82 c and84 c) or by rotating a groove about its vertex. For example, changingthe angle of the primary groove face 50 will either increase or decreaseall of the dihedral 1-2 and dihedral 1-3 errors along a given lamina.This contrasts to changes in inclination where the dihedral 1-2 anddihedral 1-3 errors can be varied differently in each groove along agiven lamina. Similarly, the half angle for the side grooves may vary,resulting in a corresponding change in dihedral 2-3. Note that for sidegrooves that are orthogonal or nearly orthogonal (within about 1°) tothe primary groove face, dihedral 1-2 and dihedral 1-3 are veryinsensitive to changes in side groove half angle. As a result, varyingthe half angles of the primary or side grooves during machining will notallow dihedral 1-2 and dihedral 1-3 to vary in opposition within a givencube corner. Varying the half angles of the primary or side groovesduring machining may be used in combination with skew and/or inclinationto provide the broadest possible control over cube corner dihedral angleerrors with a minimum number of tool changes. While the magnitude of anyone of half angle errors, skew, or inclination can ranges up to about1°, cumulatively for any given cube the resulting dihedral angle erroris no more than about 1°.

[0109] For simplicity during fabrication, skew and/or inclination arepreferably introduced such that the dihedral angle errors are arrangedin patterns. Preferably, the pattern comprises dihedral angle errors 1-2and 1-3 that are varied in opposition within a given cube corner.

[0110] Spot diagrams are one useful method based on geometric optics ofillustrating the spread in the retroreflected light resulting fromnon-orthogonality from a cube corner array. Cube corners are known tosplit the incoming light ray into up to six distinct return spotsassociated with the six possible sequences for a ray to reflect from thethree cube faces. The radial spread of the return spots from the sourcebeam as well as the circumferential position about the source beam maybe calculated once the three cube dihedral errors are defined (see e.g.Eckhardt, “Simple Model of Cube Corner Reflection”, Applied Optics, V10,N7, July 1971). Radial spread of the return spots is related toobservation angle while circumferential position of the return spots isrelated to presentation angle as further described in US Federal TestMethod Standard 370 (Mar. 1, 1977). A non-orthogonal cube corner can bedefined by the surface normal vectors of its three faces. Return spotpositions are determined by sequentially tracking a ray as it strikesand reflects from each of the three cubes faces. If the refractive indexof the cube material is greater than 1, then refraction in and out ofthe front surface cube must also be taken into account. Numerous authorshave described the equations related to front surface reflection andrefraction (e.g. Hecht and Zajac, “Optics”, 2^(nd) edition, AddisonWesley 1987). Note that spot diagrams are based on geometric optics andhence neglect diffraction. Accordingly, cube size and shape is notconsidered in spot diagrams.

[0111] The return spot diagram for five different cubes that arebackward canted by 7.47 degrees (e.g. FIG. 12) with errors in theprimary groove half angle of five consecutive grooves of +2, +4, +6, +8,and +10 arc minutes is depicted in FIG. 24. The half angle errors forthe side grooves are zero (dihedral 2-3=0) in this example, as are skewand inclination. All the side groove included angles are 90°. Thevertical and horizontal axes in FIG. 24 correspond to reference planes28 and 24, respectively. Note that changes in the slope of the primarygroove face result in return spots located primarily along the verticaland horizontal axes.

[0112] The dihedral errors as a function of primary groove half angleerrors are presented in Table 3 for the same errors used to produce FIG.24. Note that dihedral 1-2 and dihedral 1-3 have the same magnitude andsign and thus, do not vary in opposition. TABLE 3 Primary Groove ErrorDihedral 1-2 Dihedral 2-3 Dihedral 1-3 (arc minutes) (arc minutes) (arcminutes) (arc minutes) 2 1.4 0.0 1.4 4 2.8 0.0 2.8 6 4.2 0.0 4.2 8 5.70.0 5.7 10 7.1 0.0 7.1

[0113] The return spot diagram for the same type of backward cantedcubes with dihedral 2-3 errors of −20, −15, −10, −5, and 0 arc minutesis depicted in FIG. 25. The half angle errors for the primary groove arezero (dihedral 1-3=dihedral 1-2=0) in this example, as are skew andinclination. As stated previously, variations in the half angles for theside grooves may be used to produce the dihedral 2-3 errors. Thedihedral 2-3 errors result in return spots located primarily along thehorizontal axis.

[0114]FIG. 26 depicts a return spot diagram resulting from combiningprimary groove half angle errors with variations in the half angles forthe side grooves for the same type of backward canted cubes as describedwith reference to FIGS. 24-25. In this example, the primary groove halfangle error is 10 arc minutes while dihedral 2-3 error is 0, 2, 4, and 6arc minutes respectively for four different adjacent cubes on thelamina. A constant included angle error of +3 arc minutes could be usedto produce these side grooves, with the opposing half angle errors asshown in Table 4. The return spots are again located primarily along thevertical and horizontal axes, with some spreading in the horizontalplane due to the nonzero values for dihedral 2-3. Overall the returnspot diagram is localized and non-uniform.

[0115] The dihedral errors as a function of primary groove half angleerrors are presented in Table 4 for the errors used to produce FIG. 26.Note that dihedral 1-2 and dihedral 1-3 have the same magnitude and signand hence do not vary in opposition (i.e. are substantially free ofvarying in opposition). Note that a given cube corner is formed by twoadjacent side grooves and preferably a primary groove surface. The upperside groove in FIG. 22 forms dihedral 1-3 while the lower side grooveforms dihedral 1-2. The intersection of the upper and lower side groovesforms dihedral 2-3. Side groove included angle is the sum of the upperand lower half angle errors for a groove that forms adjacent cubes (e.g.with reference to Table 4 the total included angle is +3 arc minutes andresults from adding the upper half angle of a first cube with the lowerhalf angle of the adjacent cube). TABLE 4 Dihedral Dihedral DihedralLower Upper Half 1-2 2-3 1-3 HalfAngle Angle Cube (arc (arc (arc ErrorError No. minutes) minutes) minutes) (arc minutes) (arc minutes) 1 7.14.0 7.1 3 1 2 7.1 6.0 7.1 2 4 3 7.1 2.0 7.1 −1 3 4 7.1 0.0 7.1 0 0

[0116] The preceding examples (i.e. FIGS. 24-26) were for backwardcanted cubes with varying half angle errors. In an analogous manner,forward canted cubes can be shown to have qualitatively similar returnspot diagramss, i.e. substantially non-uniform with spots localizedespecially along the horizontal and vertical axes. Dihedral 1-2 anddihedral 1-3 of forward canted cubes also will have the same magnitudeand sign and thus are substantially free of varying in opposition. Inconsideration of the uses of cube corner retroreflective sheeting, it isapparent that localized, non-uniform spot diagramss (e.g. FIGS. 24-26)are generally undesirable. Sheeting may be placed on signs in a widevariety of orientations, both as the background color as well as in somecases as cut out letters. Furthermore, signs may typically be positionedon the right, on the left, or above the road. In the case of vehiclesmarked with retroreflective sheeting for conspicuity, the position ofthe vehicle relative to the viewer is constantly changing. Both the leftand right headlights of a vehicle illuminate a retroreflective target,and the position of the driver is quite different with respect to theseheadlights (differing observation and presentation angles). Vehiclessuch as motorcycles, where the driver is directly above the headlight,are commonly used. Finally, all of the relevant angles defining theviewing geometry change with distance of the driver/observer to theretroreflective sheeting or target. All of these factors make it clearthat a relatively uniform spread of return spots is highly desirable inretroreflective sheeting. Because of the flexibility to easily introducea wide range of dihedral angle errors, including dihedral 1-2 anddihedral 1-3 that vary in opposition, skew and/or inclination may beutilized to provide a relatively uniform spot return diagram.

[0117]FIG. 27 presents a return spot diagram resulting from variationsin only inclination on a single lamina with the same backward cantedcubes used in FIGS. 24-26. Half angle errors for the side grooves are+1.5 arc minutes on each side (dihedral 2-3 and side groove angle errorof +3 arc minutes) and primary groove error is zero. Skew is constant inthis example at +7 arc minutes. Inclination is varied in a repeatingpattern over every four grooves (i.e. two grooves +5 arc minutes, thentwo grooves −1 arc minute). The spot pattern is much more uniformlydistributed both radially (observation) and circumferentially(presentation) in comparison with FIGS. 24-26.

[0118] The dihedral errors for this example of varying inclination arepresented in Table 5. The order of machining of the inclinations (arcminutes) is −1, +5, +5, −1 in a repeating pattern. For example withreference to cube no. 1, the first side groove has an inclination of −1and the second side groove has an inclination of +5. Note that dihedral1-2 and dihedral 1-3 vary in opposition with different magnitudes(absolute value of the dihedral angle errors are unequal) and signs.TABLE 5 Inclination Dihedral 1-2 Dihedral 3-2 Dihedral 1-3 Cube No. (arcminutes) (arc minutes (arc minutes) (arc minutes) 1 −1, 5 5.1 3.0 −7.9 2  5, 5 0.8 3.0 −7.9 3   5, −1 0.8 3.0 −3.7 4 −1, −1 5.1 3.0 −3.7

[0119]FIG. 28 depicts the return spot diagram resulting from the samegeometry as FIG. 27, except skew is −7 arc minutes instead of +7 arcminutes for all side grooves. The spot diagram is again uniformlydistributed in comparison with FIGS. 24-26 as well as complementary tothe spot diagram shown in FIG. 27. The dihedral errors for this exampleof varying inclination are presented in Table 6. Note once again thatdihedral 1-2 and dihedral 1-3 vary in opposition, differing both inmagnitude and/or sign. The change in sign of the skew has resulted in aswitch in the magnitude and sign of dihedral 1-2 and 1-3 in comparisonto Table 5. TABLE 6 Inclination Dihedral 1-2 Dihedral 3-2 Dihedral 1-3(arc minutes) (arc minutes) (arc minutes) (arc minutes) −1, 5 −3.7 3.00.8   5, 5 −7.9 3.0 0.8   5, −1 −7.9 3.0 5.1 −1, −1 −3.7 3.0 5.1

[0120] The positive and negative skews of the two preceding examples maybe combined, providing the spot diagram of FIG. 29. This combinationmight be achieved by machining half of the lamina with +7 arc minutes ofskew and the other half with −7 arc minutes of skew. Alternatively, thepositive and negative skew could be combined within each lamina,resulting in both skew and inclination being varied concurrently withina given lamina. In the latter case, a small number of other return spotswould result from the cubes positioned at the boundary of the positiveand negative skew sections. The spot diagram is particularly uniformlydistributed in comparison with FIGS. 24-26 as it results from thecombination of the spot diagrams in FIGS. 27 and 28. A combination ofthe dihedral errors as shown in Tables 5 and 6 are associated with thisspot diagram, with dihedral 1-2 and dihedral 1-3 differing in magnitudeand sign, varying in opposition.

[0121]FIG. 30 presents the same half angle errors, skews, andinclinations as FIG. 29 except for cubes that are forward canted by7.47°. The spot diagram is also uniformly distributed although slightlydifferent than the backward canted spot diagram of FIG. 29. The dihedralerrors associated with this spot diagram are summarized in Table 7,where dihedral 1-2 and dihedral 1-3 again vary in opposition, includingat least one cube where dihedral 1-2 and dihedral 1-3 differ inmagnitude and/or sign. TABLE 7 Inclination Skew Dihedral 1-2 Dihedral3-2 Dihedral 1-3 (arc minutes) (arc minutes) (arc minutes) (arc minutes)(arc minutes) −1, 5 7 4.3 3.0 −7.2   5, 5 7 0.1 3.0 −7.2   5, −1 7 0.13.0 −2.9 −1, −1 7 4.3 3.0 −2.9 −1, 5 −7 −2.9 3.0 0.1   5, 5 −7 −7.2 3.00.1   5, −1 −7 −7.2 3.0 4.3 −1, −1 −7 −2.9 3.0 4.3

[0122] The same skew and inclination combinations may also be utilizedadvantageously in combination with sideways canted cube corners toprovide a uniformly distributed spot diagram. Sideways canted cubes, aspreviously discussed, comprise two different cube orientations withinthe same row. Preferably, care should be taken to apply the combinationsof skew and/or inclination equally to both types of cube in a given row(e.g. alpha (α) and beta (β)) in order to obtain uniform performance atvarious entrance and orientation angle combinations. The return spotdiagram for cubes that are sideways canted by 6.03° (FIG. 6, alignmentangle 90° or 270°) utilizing skew and inclination is shown in FIG. 31.The same combinations of +7 and −7 arc minutes of skew with −1 and 5 arcminutes of inclination were applied equally to both the alpha (α) andbeta (β) cubes. Half angle errors for the side grooves are +1.5 arcminutes on each side (dihedral 2-3 and side groove angle error of +3 arcminutes) and primary groove error is zero. The spot diagram is veryuniformly distributed in observation and presentation angle. Thedihedral errors associated with this spot diagram are summarized inTable 8, where dihedral 1-2 and dihedral 1-3 again vary in opposition,including at least one cube where dihedral 1-2 and dihedral 1-3 differin magnitude and/or sign. TABLE 8 Dihedral Dihedral Dihedral SkewInclination Inclination 1-2 3-2 1-3 Lower Upper (arc (arc (arc (arc (arc(arc Included Included minutes) minutes) minutes) minutes) minutes)minutes) angle (°) angle (°) 7 −1 −1 4.3 3.0 −3.9 52.387 37.613 7 −1 55.1 3.0 −7.4 37.613 52.387 7 5 5 −0.5 3.0 −7.6 52.387 37.613 7 5 −1 1.53.0 −2.7 37.613 52.387 7 −1 5 4.3 3.0 −7.6 52.387 37.613 7 5 5 1.5 3.0−7.4 37.613 52.387 7 5 −1 −0.5 3.0 −3.9 52.387 37.613 7 −1 −1 5.1 3.0−2.7 37.613 52.387 −7 −1 −1 −3.9 3.0 4.3 37.613 52.387 −7 −1 5 −2.7 3.01.5 52.387 37.613 −7 5 5 −7.6 3.0 −0.5 37.613 52.387 −7 5 −1 −7.4 3.05.1 52.387 37.613 −7 −1 5 −3.9 3.0 −0.5 37.613 52.387 −7 5 5 −7.4 3.01.5 52.387 37.613 −7 5 −1 −7.6 3.0 4.3 37.613 52.387 −7 −1 −1 −2.7 3.05.1 52.387 37.613

[0123] A characteristic of the exemplary cube corner elements of Tables5-8 is the formation of at least one and typically a plurality of PGcube corner elements in a row having three dihedral angle errors whereinthe dihedral angle errors are different from each other. Anothercharacteristic is that the dihedral angle errors, and thus the skewand/or inclination, is arranged in a repeating pattern throughout alamina or row of adjacent cube corner elements. Further the adjacentlamina or row is preferably optically identical except rotated 180°about the z-axis forming pairs of laminae or pairs of rows.

[0124] Methods of machining laminae and forming a master tool comprisinga plurality of laminae is known, such as described in U.S. Pat. No.6,257,860 (Lutrell et al.). For embodiments wherein the side grooves aresubstantially free of skew and/or inclination, side grooves may beformed in a plurality of stacked laminae, such as described in U.S. Pat.No. 6,257,860 (Lutrell et al.) and U.S. Patent No. 6,159,407 (Krinke etal.).

[0125] Accordingly, further described herein are methods of machininglaminae by providing a lamina or laminae and forming V-shaped grooves onworking surface 16 of the lamina wherein the grooves are formed with anyone or combinations of the features previously described.

[0126] In general, the lamina(e) may comprise any substrate suitable forforming directly machined grooves on the edge. Suitable substratesmachine cleanly without burr formation, exhibit low ductility and lowgraininess and maintain dimensional accuracy after groove formation. Avariety of machinable plastics or metals may be utilized. Suitableplastics comprise thermoplastic or thermoset materials such as acrylicsor other materials. Machinable metals include aluminum, brass, copper,electroless nickel, and alloys thereof. Preferred metals includenon-ferrous metals. Suitable lamina material may be formed into sheetsby for example rolling, casting chemical deposition, electro-depositionor forging. Preferred machining materials are typically chosen tominimize wear of the cutting tool during formation of the grooves.

[0127] The diamond tools suitable for use are of high quality such asdiamond tools that can be purchased from K&Y Diamond (Mooers, N.Y.) orChardon Tool (Chardon, Ohio). In particular, suitable diamond tools arescratch free within 10 mils of the tip, as can be evaluated with a 2000×white light microscope. Typically, the tip of the diamond has a flatportion ranging in size from about 0.00003 inches (0.000762 mm) to about0.00005 inches (0.001270 mm). Further, the surface finish of suitablediamond tools preferably have a roughness average of less than about 3nm and a peak to valley roughness of less than about 10 nm. The surfacefinish can be evaluated by forming a test cut in a machinable substrateand evaluating the test cut with a micro-interferometer, such as can bepurchased from Wyko (Tucson, Ariz.), a division of Veeco.

[0128] The V-shaped grooves are formed with a diamond-tooling machinethat is capable of forming each groove with fine precision. MooreSpecial Tool Company, Bridgeport, Conn.; Precitech, Keene, N.H.; andAerotech Inc., Pittsburg, Pa., manufacture suitable machines for suchpurpose. Such machines typically include a laserinterferometer-positioning device. A suitable precision rotary table iscommercially available from AA Gage (Sterling Heights, Mich.); whereas asuitable micro-interferometer is commercially available from ZygoCorporation (Middlefield, Conn.) and Wyko (Tucson, Ariz.) a division ofVeeco. The precision (i.e. point to point positioning) of the groovespacing and groove depth is preferably at least as precise as +/−500 nm,more preferably at least as precise as +/−250 nm and most preferably atleast as precise as +/−100 nm. The precision of the groove angle is atleast as precise as +/−2 arc minutes (+/−0.033 degrees), more preferablyat least as precise as +/−1 arc minute (+/−0.017 degrees), even morepreferably at least at precise as +/−½ arc minute (+/−0.0083 degrees),and most preferably at least as precise as +/−¼ arc minute (+/−0.0042degrees) over the length of the cut (e.g. the thickness of the lamina).Further, the resolution (i.e. ability of groove forming machine todetect current axis position) is typically at least about 10% of theprecision. Hence, for a precision of +/−100 nm, the resolution is atleast +/−10 nm. Over short distances (e.g. about 10 adjacent parallelgrooves), the precision is approximately equal to the resolution. Inorder to consistently form a plurality of grooves of such fine accuracyover duration of time, the temperature of the process is maintainedwithin +/−0.1° C. and preferably within +/−0.01° C.

[0129] While the change in shape of a single cube corner element due toskew and/or inclination is small with respect to a single element (e.g.limited primarily to changes in the dihedral angles), it is evident thatforming skewed and/or inclined grooves in a stack of laminae may beproblematic. Since the side grooves deviate from parallel up to as muchas 1°, significantly varying cube geometries may be produced across thestack. These variations increase as the stack size increases. Thecalculated maximum number of laminae that can be machined concurrently(i.e. in a stack) without creating significantly varying cube geometriesis as few as two laminae (e.g. for 1° skew, 0.020 inch (0.508 mm) thicklamina with 0.002 inch (0.0508 mm) side groove spacing).

[0130] Due to the problems of machining stacks of laminae having skewedand/or inclined side grooves, in the practice of such embodiments theside grooves are preferably formed in individual laminae with agroove-forming machine. A preferred method for forming grooves on theedge portion of individual laminae, assembling the laminae into a mastertool, and replicating the microstructured surface of the assembledlaminae is described in U.S. patent application Ser. No. 10/383039,entitled “Methods of Making Microstructured Lamina and Apparatus” filedMar. 6, 2003, incorporated herein by reference. U.S. patent applicationSer. No. 10/383039 was concurrently filed with Provisional PatentApplication Serial No. 60/452464, to which the present applicationclaims priority.

[0131] To form a master tool of suitable size for formingretroreflective sheeting, a plurality of toolings (also referred to astiles) are formed by electroplating the surface of the master tool toform negative copies, subsequently electroplating the negative copies toform positive copies, electroplating the positive copies to form asecond generation negative copies, etc. The positive copy has the samecube corner element structure as the master tool, whereas the negativecopy is the cube cavity replica. Accordingly, a negative copy tool isemployed to make a positive copy (i.e. cube corner element) sheetingwhereas, a positive copy tool is employed to make a negative copy (i.e.cube corner cavity) sheeting. Further, retroreflective sheeting maycomprise combination of cube corner elements and cube corner cavitymicrostructures. Electroforming techniques such as described in U.S.Pat. Nos. 4,478,769 and 5,156,863 (Pricone) as well as U.S. Pat. No.6,159,407 (Krinke) are known. Tiling such toolings together can thenassemble a master tool of the desired size. In the present invention thetoolings are typically tiled in the same orientation.

[0132] As used herein, “sheeting” refers to a thin piece of polymeric(e.g. synthetic) material upon which cube corner microstructures havebeen formed. The sheeting may be of any width and length, such dimensiononly being limited by the equipment (e.g. width of the tool, width ofthe slot die orifice, etc.) from which the sheeting was made. Thethickness of retroreflective sheeting typically ranges from about 0.004inches (0.1016 mm) to about 0.10 inches (2.54 mm). Preferably thethickness of retroreflective sheeting is less than about 0.020 inches(0.508 mm) and more preferably less than about 0.014 inches (0.3556 mm).The retroreflective sheeting may further include surface layers such asseal films or overlays. In the case of retroreflective sheeting, thewidth is typically at least 30 inches (122 cm) and preferably at least48 inches (76 cm). The sheeting is typically continuous in its lengthfor up to about 50 yards (45.5 m) to 100 yards (91 m) such that thesheeting is provided in a conveniently handled roll-good. Alternatively,however, the sheeting may be manufactured as individual sheets ratherthan as a roll-good. In such embodiments, the sheets preferablycorrespond in dimensions to the finished article. For example, theretroreflective sheeting, may have the dimensions of a standard U.S.sign (e.g. 30 inches by 30 inches (76 cm by 76 cm) and thus themicrostructured tool employed to prepare the sheeting may have about thesame dimensions. Smaller articles such as license plates or reflectivebuttons may employ sheeting having correspondingly smaller dimensions.

[0133] The retroreflective sheet is preferably manufactured as anintegral material, i.e. wherein the cube-corner elements areinterconnected in a continuous layer throughout the dimension of themold, the individual elements and connections therebetween comprisingthe same material. The surface of the sheeting opposing themicroprismatic surface is typically smooth and planar, also beingreferred to as the “land layer”. The thickness of the land layer (i.e.the thickness excluding that portion resulting from the replicatedmicrostructure) is between 0.001 and 0.100 inches and preferably between0.003 and 0.010 inches. Manufacture of such sheeting is typicallyachieved by casting a fluid resin composition onto the tool and allowingthe composition to harden to form a sheet. A preferred method forcasting fluid resin onto the tool is described in U.S. patentapplication Ser. No. 10/382375, entitled “Method of MakingRetroreflective Sheeting and Slot Die Apparatus”, filed Mar. 6, 2003,incorporated herein by reference. U.S. patent application Ser. No.10/382375 was concurrently filed with Provisional Patent ApplicationSerial No. 60/452464, to which the present invention claims priority.

[0134] Optionally, however, the tool can be employed as an embossingtool to form retroreflective articles, such as described in U.S. Pat.No. 4,601,861 (Pricone). Alternatively, the retroreflective sheeting canbe manufactured as a layered product by casting the cube-corner elementsagainst a preformed film as taught in PCT application No. WO 95/11464and U.S. Pat. No. 3,684,348, or by laminating a preformed film topreformed cube-corner elements. In doing so the individual cube-cornerelements are interconnected by the preformed film. Further, the elementsand film are typically comprised of different materials.

[0135] In the manufacture of the retroreflective sheeting, it ispreferred that the channels of the tool are roughly aligned with thedirection of the advancing tool as further described in U.S. patentapplication Ser. No. 60/452605, entitled “Methods of MakingRetroreflective Sheeting and Articles”, filed Mar. 6, 2003. U.S. patentapplication Ser. No. 60/452605 was filed concurrently with ProvisionalPatent Application Serial No. 60/452464, to which the present inventionclaims priority. Accordingly, prior to any further manufacturing steps,the primary groove of the sheeting would be substantially parallel tothe edge of the roll of the sheeting. The present inventors have foundthat orienting the channels in this downweb manner allows for fasterreplication than when the primary groove is oriented cross web. It issurmised that the primary groove and other cube structures combine toform channels for improved resin flow.

[0136] Suitable resin compositions for the retroreflective sheeting ofthis invention are preferably transparent materials that aredimensionally stable, durable, weatherable, and readily formable intothe desired configuration. Examples of suitable materials includeacrylics, which have an index of refraction of about 1.5, such asPlexiglas brand resin manufactured by Rohm and Haas Company;polycarbonates, which have an index of refraction of about 1.59;reactive materials such as thermoset acrylates and epoxy acrylates;polyethylene based ionomers, such as those marketed under the brand nameof SURLYN by E. I. Dupont de Nemours and Co., Inc.;(poly)ethylene-co-acrylic acid; polyesters; polyurethanes; and celluloseacetate butyrates. Polycarbonates are particularly suitable because oftheir toughness and relatively higher refractive index, which generallycontributes to improved retroreflective performance over a wider rangeof entrance angles. These materials may also include dyes, colorants,pigments, UV stabilizers, or other additives.

[0137] A specular reflective coating such as a metallic coating can beplaced on the backside of the cube-corner elements. The metallic coatingcan be applied by known techniques such as vapor depositing orchemically depositing a metal such as aluminum, silver, or nickel. Aprimer layer may be applied to the backside of the cube-corner elementsto promote the adherence of the metallic coating. In addition to or inlieu of a metallic coating, a seal film can be applied to the backsideof the cube-corner elements; see, for example, U.S. Pat. Nos. 4,025,159and 5,117,304. The seal film maintains an air interface at the backsideof the cubes that enables total internal reflection at the interface andinhibits the entry of contaminants such as soil and/or moisture. Furthera separate overlay film may be utilized on the viewing surface of thesheeting for improved (e.g. outdoor) durability or to provide an imagereceptive surface. Indicative of such outdoor durability is maintainingsufficient brightness specifications such as called out in ASTMD49560-1a after extended durations of weathering (e.g. 1 year, 3 years).Further the CAP Y whiteness is preferably greater than 30 before andafter weathering.

[0138] An adhesive layer also can be disposed behind the cube-cornerelements or the seal film to enable the cube-corner retroreflectivesheeting to be secured to a substrate. Suitable substrates include wood,aluminum sheeting, galvanized steel, polymeric materials such aspolymethyl methacrylates, polyesters, polyamids, polyvinyl fluorides,polycarbonates, polyvinyl chlorides, polyurethanes, and a wide varietyof laminates made from these and other materials.

[0139] With reference to FIG. 6, the laminae are preferably alignedvertically. In doing so, upon replication a row of elements is derivedfrom each lamina. Alternatively, however, these same optical featuresmay be derived from horizontally aligned laminae. The common plane thata face of each element within a row share to within about 3-4 micronsmay vary slightly (e.g. less than 1°) for horizontally aligned laminae.It is evident that a row of cubes was derived from a lamina due to thepresence of slight vertical or horizontal misalignments as can beobserved with, for example, scanning electron microscopy.

[0140] Regardless of the method of making the retroreflective sheetingor whether the master tool was derived from a lamina technique or othertechnique, the sheeting of the invention has certain unique opticalfeatures that can be detected by viewing the sheeting with a microscopeor interferometer as previously described.

[0141] In one aspect, the retroreflective sheeting comprises a row ofcube corner elements or an array of cube corner element wherein theincluded angle between a first and second concurrent element in a rowdiffers from the included angle between a second and a third concurrentelement in the row. With respect to the sheeting, the row is defined bythe elements wherein a face of each element within the row shares acommon plane (e.g. primary groove face, working surface 12 or 14). Themagnitude of the difference in included angle between adjacent cubes aswell as other preferred characteristics (e.g. arranged in a repeatingpattern, common peak height, bisector planes that range form beingmutually nominally parallel to non-parallel by less than 1°) within arow or array is the same as previous described with respect to thelamina.

[0142] Alternatively or in combination thereof, the retroreflectivesheeting comprises a row or an array of cube corner elements (e.g. PGcube corner elements) wherein at least a portion of the elements in arow or array are predominantly sideways canted, the elements having analignment angles between 45° and 135° and/or having an alignment anglebetween 225° and 315° relative to the dihedral edge that issubstantially perpendicular to a row of elements in plan view. Inpreferred embodiments, the retroreflective sheeting comprises a row ofcube corner elements or an array having cube corner elements having eachof these alignment angles. Such array is substantially free ofpredominantly forward canted or predominantly backward canted cubecorner elements. The retroreflective sheeting comprising predominantlysideways canted cube corner elements may further comprise any of thecharacteristics previously described with regard to the lamina.

[0143] Alternatively or in combination thereof, the retroreflectivesheeting comprises skewed and/or inclined grooves. Hence, the row or thearray wherein at least two adjacent grooves and preferably all thegrooves of the (e.g. side) groove set are non-parallel by amount rangingfrom greater than nominally parallel to about 1° and may further includethe various attributes described with regard to lamina comprising thisfeature.

[0144] In another aspect, alone or in combination with differingincluded angles and/or sideways canting, the retroreflective sheetingmay comprise a row or elements or an array wherein the grooves of theside groove set are nominally parallel to each other, yet range frombeing nominally parallel to non-parallel to reference plane 28.

[0145] The retroreflective sheeting is useful for a variety of uses suchas traffic signs, pavement markings, vehicle markings and personalsafety articles, in view of its high retroreflected brightness. Thecoefficient of retroreflection, R_(A), may be measured according to USFederal Test Method Standard 370 at −4° entrance, 0° orientation, atvarious observation angles. The resulting sheeting meets brightnessspecifications called out in ASTM D4956-1a “The Standard Specificationfor Retroreflective Sheeting for Traffic Control” for Type IX sheeting.Additionally, specified brightness minimums are significantly exceededfor −4° entrance, an average of 0° and 90° orientation, 0° presentationand various observation angles. The brightness is preferably at least625 candelas per lux per square meter (CPL), more preferably at least650 CPL, even more preferably at least 675 CPL, and most preferably atleast 700 CPL at an observation angle of 0.2°. Alternatively, andpreferably in addition thereto, the brightness at an observation angleof 0.33° is preferably at least 575 CPL, more preferably at least 600CPL, even more preferably at least 625 CPL, and most preferably at least650 CPL. In addition or in the alternative, the brightness at anobservation angle of 0.5° is preferably at least 375 CPL, morepreferably at least 400 CPL, even more preferably at least 425 CPL, andmost preferably at least 450 CPL. Further, the brightness at anobservation angle of 1.0° is preferably at least 80 CPL, more preferablyat least 100 CPL, and most preferably at least 120 CPL. Likewise, thebrightness at an observation angle of 1.5° is preferably at least 20 CPLand more preferably at least 25 CPL. The retroreflective sheeting maycomprise any combination of brightness criteria just stated.

[0146] Improved brightness in the region around 0.5 observation angle(i.e. 0.4 to 0.6) is particularly important for viewing traffic signs(e.g. right should mounted) from passenger vehicles at distances ofroughly 200 to 400 feet and for the viewing of traffic signs (e.g. rightshould mounted) from drivers of large trucks at distances of about 450to 950 feet.

[0147] Objects and advantages of the invention are further illustratedby the following examples, but the particular materials and amountsthereof recited in the examples, as well as other conditions anddetails, should not be construed to unduly limit the invention.

EXAMPLES 1A and 1B

[0148] Grooves were formed in individual lamina, the individual laminaassembled, and the microstructured surface replicated as described inpreviously cited U.S. patent application Ser. No. 10/383039, filed Mar.6, 2003. U.S. patent application Ser. No. 10/383039 was filedconcurrently with Provisional Patent Application Serial No. 60/452464 towhich the present application claims priority. All the machined laminaehad the geometry depicted in FIGS. 6 and 7, with slight variations dueto varying the half angle error, skew and inclination of the sidegrooves. The lamina thickness was 0.0075 inches (0.1905 mm) and the sidegroove spacing was 0.005625 inches (0.1428 mm) except for the slightvariations just described. A repeating pattern of eight cubes wassequentially formed on each lamina. This repeating pattern of cubes wasformed by varying the half angle errors, skew, and inclination of theside grooves as set forth in forthcoming Tables 10-14. Each row in thetables defines the parameters used during machining of an individualside groove. The cube corner dihedral errors, as defined in FIG. 22, areformed by the two adjacent side grooves that intersect the primarygroove surface to form each cube. Hence, the rows defining dihedralangle errors are offset in the table to clarify their adjacent sidegrooves.

[0149] Eight laminae that differed with regard to the angle error and/orskew and/or inclination of the side grooves were formed such that thedihedral angle errors reported in each of the following Tables 10-14were obtained with the exception of Table 13 wherein the skew of aportion of the side grooves was modified.

[0150] Lamina 1 and Lamina 2

[0151] The side groove parameters of the first lamina as well as thesecond lamina, the second lamina being an opposing lamina to the firstlamina, are reported in Tables 10 and 11, respectively. The primarygroove half angle error was −8 arc minutes for all the primary grooves.Side groove nominal included angles (the angles required to produceorthogonal cubes) were 75.226° and 104.774°. The included angle errorfor all side grooves was −9.2 arc minutes, resulting in actual sidegroove included angles of 75.073° and 104.621°. While the included angleerror was constant for the side grooves, the half angle errors werevaried. Half angle errors for the first lamina type ranged from −14.8arc minutes to 5.6 arc minutes as shown in column 3 of Table 10. Thehalf angle errors are presented in groups of two (totaling −9.2 arcminutes) corresponding to the two half angles for each side groove. Thedihedral 2-3 error results from the combination of half angle errors onadjacent side grooves and is summarized in column 4. Dihedral 2-3 errorsvaried from −1.6 arc minutes to −16.8 arc minutes for the first lamina.

[0152] Skew and inclination are set forth in columns five and six ofTable 10, respectively. Skew ranged from −8.0 arc minutes to 15.0 arcminutes for the first lamina. Inclination varied from −6.1 arc minutesto 10.8 arc minutes. The 1-2 and 1-3 dihedral errors resulting from skewand inclination of the side grooves are shown in the final two columns.Note that dihedral errors 1-2 and 1-3 varied in opposition, with atleast one cube in the lamina comprising dihedral errors 1-2 and 1-3 withdifferent magnitudes and/or signs.

[0153] The side grooves of the second lamina, is summarized in Table 11and is closely related to the lamina of Table 10. The first and secondcolumns, that set forth the nominal side groove angle as well as sidegroove included angle error, are identical. All other columns for sidegroove parameters (half angle errors, skew and inclination) as well asdihedral angle errors are inverted in relation to Table 10. Thisreflects the fact that an opposing lamina is optically identical to itscounterpart except rotated 180° about the z-axis.

[0154] Lamina 4, Lamina 6 and Lamina 8

[0155] For simplicity, the side groove parameters of the fourth, sixth,and eight lamina that are respectively opposing the third, fifth andseventh lamina are not reiterated since the side grooves parameter havethis same inverted relationship as just described.

[0156] Lamina 3

[0157] The side groove parameter of the third lamina is set forth inTable 12. Primary groove half angle error was −8 arc minutes. The basicgeometry (dimensions and nominal side groove included angles) was thesame as the first lamina type. The actual included angle error for allside grooves was again −9.2 arc minutes. Half angle errors for thesecond lamina type side grooves ranged from −14.8 arc minutes to 5.6 arcminutes. Dihedral 2-3 errors varied from −1.6 arc minutes to −16.8 arcminutes. Skew ranged from −14.0 arc minutes to 21.3 arc minutes whileinclination varied from −12.7 arc minutes to 16.8 arc minutes for thislamina type. The 1-2 and 1-3 dihedral errors (shown in the final twocolumns) varied in opposition.

[0158] Lamina 5

[0159] The groove parameters of the fifth lamina is set forth in Table13. The primary groove half angle error was −4 arc minutes. The basicgeometry (dimensions and nominal side groove included angles) was thesame as the preceding laminae. The included angle error for all sidegrooves was −1.6 arc minutes, resulting in actual side groove includedangles of 75.199° and 104.747°. Half angle errors for the third laminatype ranged from −5.2 arc minutes to 3.6 arc minutes. Dihedral 2-3errors varied from −7.2 arc minutes to 4.0 arc minutes. Skew ranged from−7.0 arc minutes to 9.5 arc minutes while inclination varied from −8.2arc minutes to 1.4 arc minutes. The 1-2 and 1-3 dihedral errors (shownin the final two columns) varied in opposition.

[0160] Lamina 7

[0161] The side groove parameter for the seventh lamina is set forth inTable 14. The primary groove half angle error was −4.0 arc minutes. Thebasic geometry (dimensions and nominal side groove included angles) wasthe same as the first lamina type. The actual included angle error forall side grooves was again −1.6 arc minutes. Half angle errors rangedfrom −5.2 arc minutes to 3.6 arc minutes. Dihedral 2-3 errors variedfrom −7.2 arc minutes to 4.0 arc minutes. Skew ranged from −5.3 arcminutes to 5.3 arc minutes while inclination varied from −2.1 arcminutes to 4.6 arc minutes for this lamina type. The 1-2 and 1-3dihedral errors (shown in the final two columns) varied in opposition.

[0162] A total of 208 laminae were assembled such that the non-dihedraledges of the elements of opposing laminae contacted each other to aprecision such that the assembly was substantially free of verticalwalls (e.g. walls greater than 0.0001 in lateral dimensions). Thelaminae were assembled such that the lamina order 1-8 was sequentiallyrepeated throughout the assembly and the structured surface of theassembly was then replicated by electroforming to create a cube cavitytool. The assembly and electroforming process is further described inpreviously cited U.S. patent application Ser. No. 10/383039, filed Mar.6, 2003. U.S. patent application Ser. No. 10/383039 was filedconcurrently with Provisional Patent Application Serial No. 60/452464 towhich the present application claims priority.

[0163] For Example 1A, the tool was used in a compression molding presswith the pressing performed at a temperature of approximately 375° F.(191° C.) to 385° F. (196° C.), a pressure of approximately 1600 psi,and a dwell time of 20 seconds. The molded polycarbonate was then cooledto about 200° F. (100° C.) over 5 minutes.

[0164] For Example 2A, molten polycarbonate was cast onto the toolsurface as described in previously cited U.S. patent application Ser.No. 10/382375, filed Mar. 6, 2003. U.S. patent application Ser. No.10/382375 was filed concurrently with Provisional Patent ApplicationSerial No. 60/452464 to which the present application claims priority.

[0165] For both Example 1A and 1B, a dual layer seal film comprising 0.7mils polyester and 0.85 mils amorphous copolyester was applied to thebackside of the cube-corner elements by contacting the amorphouscopolyester containing surface to the microstructured polycarbonate filmsurface in a continuous sealing process. The construction was passedcontinuously through a rubber nip roll having a Teflon sleeve and aheated steel roll. The surface of the rubber nip roll was about 165° F.and the surface of the heated steel roll was about 405° F. The nippressure was about 70 pounds/per linear inch and speed was 20 feet perminute. Brightness retention after sealing was about 70%.

[0166] The resulting sheeting meets brightness specifications called outin ASTM D4956-1a “The Standard Specification for RetroreflectiveSheeting for Traffic Control” for Type IX sheeting. Additionally,specified brightness minimums are significantly exceeded for −4°entrance, an average of 0° and 90° orientation, 0° presentation andvarious observation angles as follows: TABLE 9 Example 1A ExampleComparative Comparative Compression 1B Retroreflective RetroreflectiveMolded Extrusion Sheeting 2 Sheeting 3 Sheeting Sheeting Observation Avg0/90 Avg 0/90 Avg 0/90 Avg 0/90 Angle CPL CPL CPL CPL 0.2 726 489 788740 0.33 660 432 748 700 0.5 276 348 554 502 1 37 106 141 162 1.5 13 2432 35

[0167] Table 9 shows that the retroreflective sheeting of the presentinvention has a higher brightness at each of the indicated observationangles in comparison to Comparative Retroreflective Sheeting 2 andComparative Retroreflective Sheeting 3. The improved brightness in theregion around 0.5 observation angle is particularly important forviewing traffic signs (e.g. right should mounted) from passengervehicles at distances of roughly 200 to 400 feet and for the viewing oftraffic signs (e.g. right should mounted) from drivers of large trucksat distances of about 450 to 950 feet.

[0168] The sheeting of Example 1A was found to have a measureduniformity index of 2.04 for total light return within 2.0° observation.

[0169] Various modifications and alterations of this invention willbecome apparent to those skilled in the art without departing from thescope and spirit of this invention. TABLE 10 Nominal Side Groove SideGroove Side Groove Dihedral Dihedral Dihedral Included Incl. Angle HalfAngle 2-3 Error Skew Inclination 1-3 Error 1-2 Error Angle (Deg) Error(min) Errors (min) (min) (min) (min) (min) (min) 75.226 −9.2 −7.2 15.02.5 −2.0 −9.2 −16.1 −6.0 104.774 −9.2 −7.2 0.0 −0.4 −2.0 −9.2 −6.0 −16.075.226 −9.2 −7.2 −7.0 10.8 −2.0 −9.2 −7.0 −12.8 104.774 −9.2 −7.2 −8.03.1 −2.0 −16.8 −4.8 −5.7 75.226 −9.2 −14.8 −7.0 −6.0 5.6 −1.6 3.3 1.9104.774 −9.2 −7.2 14.7 −1.2 −2.0 −9.2 −12.7 −7.0 75.226 −9.2 −7.2 −1.02.5 −2.0 −16.8 −5.8 −4.9 104.774 −9.2 −14.8 −6.7 −6.1 5.6 −1.6 1.8 3.375.226 −9.2 −7.2 15.0 2.5 −2.0

[0170] TABLE 11 Nominal Side Groove Side Groove Side Groove DihedralDihedral Dihedral Included Incl. Angle Half Angle 2-3 Error SkewInclination 1-3 Error 1-2 Error Angle (Deg) Error (min) Errors (min)(min) (min) (min) (min) (min) 75.226 −9.2 −2.0 15.0 2.5 −7.2 −1.6 1.83.3 104.774 −9.2 5.6 −6.7 −6.1 −14.8 −16.8 −5.8 −4.9 75.226 −9.2 −2.0−1.0 2.5 −7.2 −9.2 −12.7 −7.0 104.774 −9.2 −2.0 14.7 −1.2 −7.2 −1.6 3.31.9 75.226 −9.2 5.6 −7.0 −6.0 −14.8 −16.8 −4.8 −5.7 104.774 −9.2 −2.0−8.0 3.1 −7.2 −9.2 −7.0 −12.8 75.226 −9.2 −2.0 −7.0 10.8 −7.2 −9.2 −6.0−16.0 104.774 −9.2 −2.0 0.0 −0.4 −7.2 −9.2 −16.1 −6.0 75.226 −9.2 −2.015.0 2.5 −7.2

[0171] TABLE 12 Nominal Side Groove Side Groove Side Groove DihedralDihedral Dihedral Included Incl. Angle Half Angle 2-3 Error SkewInclination 1-3 Error 1-2 Error Angle (Deg) Error (min) Errors (min)(min) (min) (min) (min) (min) 75.226 −9.2 −7.2 21.3 2.0 −2.0 −9.2 −19.8−8.7 104.774 −9.2 −7.2 0.0 3.0 −2.0 −9.2 −8.7 −19.7 75.226 −9.2 −7.2−7.2 16.8 −2.0 −9.2 −10.5 −15.4 104.774 −9.2 −7.2 −14.0 2.6 −2.0 −16.8−1.4 −1.5 75.226 −9.2 −14.8 −6.7 −12.7 5.6 −1.6 7.2 5.0 104.774 −9.2−7.2 20.5 −1.4 −2.0 −9.2 −15.4 −10.6 75.226 −9.2 −7.2 −7.0 2.0 −2.0−16.8 −1.6 −1.4 104.774 −9.2 −14.8 −6.7 −10.5 5.6 −1.6 5.3 7.7 75.226−9.2 −7.2 21.3 2.0 −2.0

[0172] TABLE 13 Nominal Side Groove Side Groove Side Groove DihedralDihedral Dihedral Included Incl. Angle Half Angle 2-3 Error SkewInclination 1-3 Error 1-2 Error Angle (Deg) Error (min) Errors (min)(min) (min) (min) (min) (min) 75.226 −1.6 0.4 2.1 −4.0 −2.0 −1.6 −1.43.3 104.774 −1.6 0.4 0.0 −8.2 −2.0 −1.6 3.3 −1.3 75.226 −1.6 0.4 −4.7−6.8 −2.0 −1.6 4.7 −1.7 104.774 −1.6 0.4 5.1 1.4 −2.0 −7.2 −6.8 −7.675.226 −1.6 −5.2 −7.0 1.0 3.6 4.0 1.5 −1.5 104.774 −1.6 0.4 0.4 −1.8−2.0 −1.6 −1.9 4.8 75.226 −1.6 0.4 9.5 −1.8 −2.0 −7.2 −7.5 −6.8 104.774−1.6 −5.2 −5.4 1.2 3.6 4.0 −1.4 1.4 75.226 −1.6 0.4 2.1 −4.0 −2.0

[0173] TABLE 14 Nominal Side Groove Side Groove Side Groove DihedralDihedral Dihedral Included Incl. Angle Half Angle 2-3 Error SkewInclination 1-3 Error 1-2 Error Angle (Deg) Error (min) Errors (min)(min) (min) (min) (min) (min) 75.226 −1.6 0.4 4.7 3.6 −2.0 −1.6 −7.7−1.5 104.774 −1.6 0.4 0.0 −2.1 −2.0 −1.6 −1.5 −7.7 75.226 −1.6 0.4 −4.73.6 −2.0 −1.6 −1.6 −6.8 104.774 −1.6 0.4 0.0 4.6 −2.0 −7.2 −6.8 −7.675.226 −1.6 −5.2 −4.7 3.5 3.6 4.0 −1.6 −1.6 104.774 −1.6 0.4 5.3 1.3−2.0 −1.6 −6.8 −1.6 75.226 −1.6 0.4 4.6 3.4 −2.0 −7.2 −7.5 −6.8 104.774−1.6 −5.2 −5.3 1.3 3.6 4.0 −1.6 −1.6 75.226 −1.6 0.4 4.7 3.6 −2.0

What is claimed is:
 1. A lamina comprising cube corner elements havingfaces formed from a side groove set wherein at least two grooves withinthe set are nonparallel by amounts ranging from greater than nominallyparallel to about 1°.
 2. The lamina of claim 1 wherein the elements arepreferred geometry cube corner elements.
 3. The lamina of claim 1wherein the side grooves are nonparallel to a reference plane
 28. 4. Thelamina comprising of claim 1 wherein the elements comprise dihedralangle errors having magnitudes between 1 arc minute and 60 arc minutes.5. The lamina of claim 4 wherein the dihedral angle errors are arrangedin a repeating pattern.
 6. The lamina of claim 1 wherein the groovescomprise skew.
 7. The lamina of claim 6 wherein a first groove comprisespositive and a second groove comprises negative skew.
 8. The lamina ofclaim 6 wherein the skew of a first groove varies in magnitude from theskew of a second groove.
 9. The lamina of claim 1 wherein the groovescomprise inclination.
 10. The lamina of claim 9 wherein a first groovecomprises a positive inclination and a second groove comprises anegative inclination.
 11. The lamina of claim 9 wherein the inclinationof a first groove varies in magnitude from the inclination of a secondgroove.
 12. The lamina of claim 1 wherein the side grooves comprise skewand inclination.
 13. The lamina of claim 1 wherein the elements eachhave a first face and the first faces define a primary groove face. 14.A master tool comprising a plurality of laminae in accordance withclaim
 1. 15. The master tool of claim 14 wherein the laminae areassembled such that cube corner elements of adjacent lamina are inopposing orientations.
 16. Positive and negative replicas of the mastertool of claim
 14. 17. The replica of claim 16 wherein the replication isretroreflective sheeting comprising an array of cube corner elements.18. The replica of claim 16 wherein the replication is retroreflectivesheeting comprising an array of cube corner cavities.
 19. A laminacomprising a row of cube corner elements having face formed from a sidegroove set wherein the grooves are nominally parallel to each other andrange from being nominally parallel to non-parallel to within 1° toreference plane
 28. 20. The lamina of claim 19 wherein the cube cornerelements are preferred geometry cube corner elements.
 21. An articlecomprising preferred geometry cube corner elements wherein at least onecube comprises a 1-2 dihedral angle error and a 1-3 dihedral angleerror; wherein the dihedral angle errors vary in opposition.
 22. Thearticle of claim 21 wherein the cube comprises dihedral angle errorshaving magnitudes between 1 arc minute and 60 arc minutes.
 23. Thearticle of claim 21 wherein the elements are in a row having dihedraledges that range from being nominally parallel to nonparallel by lessthan 1°
 24. The article of claim 21 wherein the dihedral angle errorsare varied in a repeating pattern.
 25. An article comprising at leastone preferred geometry cube corner element having three dihedral angleserrors wherein the dihedral angle errors are different from each other.26. The article of claim 25 wherein the dihedral angle errors havemagnitudes between 1 arc minute and 60 arc minutes.
 27. The article ofclaim 25 comprising a plurality of the elements in a row wherein atleast one dihedral edge of the elements range from being nominallyparallel to nonparallel by less than 1°
 28. The article of claim 27wherein the dihedral angle errors of the elements are varied in arepeating pattern.
 29. The article of claim 27 wherein the articlecomprises a lamina.
 30. The article of claim 27 wherein the elementseach have a first face and the first faces define a primary groove face.31. The article of claim 27 wherein the article comprises a tool. 32.The article of claim 27 wherein the article comprises retroreflectivesheeting.
 33. The article of claim 32 wherein the retroreflectivesheeting comprises cube corner elements.
 34. The article of claim 32wherein the retroreflective sheeting comprises cube corner cavities. 35.Retroreflective sheeting comprising a row of preferred geometry cubecorner elements having faces defined by a side groove set wherein atleast two grooves within the set are nonparallel by amounts ranging fromgreater than nominally parallel to about 1°.
 36. The retroreflectivesheeting of claim 35 wherein the elements comprise a lateral dimensionof less than about 0.020 inches.
 37. The retroreflective sheeting ofclaim 35 wherein the elements comprise a lateral dimension of less thanabout 0.010 inches.
 38. Retroreflective sheeting comprising a row ofpreferred geometry cube corner elements having face defined by a sidegroove set wherein the grooves are nominally parallel to each other andrange from being nominally parallel to non-parallel to within 1° toreference plane
 28. 39. The retroreflective sheeting of claim 38 whereinthe elements comprise a lateral dimension of less than about 0.020inches.
 40. The retroreflective sheeting of claim 38 wherein theelements comprise a lateral dimension of less than about 0.010 inches.41. Retroreflective sheeting comprising an array of preferred geometrycube corner elements that exhibits an average brightness at 0° and 90°orientation according to ASTM D4596-1a of at least 375 candelas/lux/m²for an entrance angle of −4° and an observation angle of 0.5°.
 42. Theretroreflective sheeting of claim 41 wherein the average brightness isat least 400 candelas/lux/m².
 43. The retroreflective sheeting of claim41 wherein the average brightness is at least 450 candelas/lux/m². 44.The retroreflective sheeting of claim 41 wherein the average brightnessat 0° and 90° is at least 625 candelas/lux/m² for an entrance angle of−4° and an observation angle of 0.2°.
 45. The retroreflective sheetingof claim 41 wherein the average brightness at 0° and 90° is at least 575candelas/lux/m² for an entrance angle of −4° and an observation angle of0.33°.
 46. The retroreflective sheeting of claim 41 wherein the averagebrightness at 0° and 90° is at least 80 candelas/lux/m² for an entranceangle of −4° and an observation angle of 1.0°.
 47. The retroreflectivesheeting of claim 41 wherein the average brightness at 0° and 90° is atleast 20 candelas/lux/m² for an entrance angle of −4° and an observationangle of 1.5°.
 48. The retroreflective sheeting of claim 41 wherein thesheeting further comprises a seal film.
 49. The retroreflective sheetingof claim 41 wherein the sheeting further comprises a specular reflectivecoating.